User talk:Zak Seidov

Hello, world! --Zak Seidov 05:13, 23 November 2010 (UTC)

Hello Zak! To answer your question: Yes, anyone can comment here! In case something ugly happens, you can revert to an earlier version (via the "history" tab). — M. F. Hasler 01:55, 29 November 2010 (UTC)

Early Birds in EKG A064413

Early Birds in EKG (A064413)

1. Numbers are given in order of their appearances in A064413.

2. From the first 888 EB's, 518 are even and 370 odd.

3. What is origin of two branches in the Figure?

How to upload the file in your talk page

How to upload file?

1. First go to your user's page:

in my case

http://oeis.org/wiki/User_talk:Zak_Seidov

and

2. start a new section:

click "+" in the menu (or "alt-shift-+")

3. enter any head line your wish:

e.g.,

"How to upload the file in your talk page"

4. Start to print your talk:

(e.g., repeat your head line).

5. Now go to left panel find "Toolbox" and click "Upload file"

6. Click "Browse..." and find in your computer the file you need:

e.g., Figure, .txt etc. and

7. Click "Open".

8. Print in the "Destination file" window "new name" -

if you wish to change the original title to "public" one.

9. Click "Upload file" below.

If it's OK, your file is now in the OEIS WIKI.

Note that points 5 - 9 can be done quite independently from your section

(even from your talk page I guess).

10. Now in the place you wish in the your talk you print:

File:FileName.jpg

(of course FileName = name of your "public" name of uploaded file, e.g., Even EKG.jpg and

10. After clicking "Show preview"

you may check if everything looks OK for you

including the Figure:

and then

11. Click "Save page". (I guess that you can't do this "Save page" without first "Show preview".)

12. Remember that in any moment you and

- ugly moment of WIKI - anyone else :( can edit your talk.

13. I hope that someone may explain this better than me and put

"How to upload the file" in OEIS WIKI QandA.

Even terms in EKG (A064413)

From the first 10000 terms of A064413, 5189 are even and 4811 odd.

From the first 1000 terms of A064413, 524 are even and 478 odd.

Even terms of EKG with indices <=1000: {index in EKG, term} (notice dependence a(n) is very close to linear one a(n)=n)

A138173: First 1000 terms

A138173 a(n) = smallest m such that m^3 begins with n^2.

(We might add also 0 because as 0^3 "begins' with 0^2.

In general, notice that for n=a^3 m=a^(2/3): a(1)=1, a(8)=4, a(27)=9, a(64)=16.

The mnemonic rule is: "if n is cube m is square"!

What about patterns in the Figures?

(Sub)sequences of distinct successive prime gaps

(See sequences A079007 and A079889.) — M. F. Hasler 14:40, 5 April 2013 (UTC)

(Sub)sequences of distinct successive prime gaps 
n - length of sequence of distinct successive prime gaps 
m - index of first prime of the sequence
p - prime(m) 
sn - sequences of n distinct successive prime gaps

{n,m,p,sn}
{1,2,3,{2}},
{2,1,2,{1,2}},
{3,7,17,{2,4,6}},
{4,23,83,{6,8,4,2}},
{5,30,113,{14,4,6,2,10}},
{6,94,491,{8,4,6,12,2,18}},
{7,219,1367,{6,8,18,10,14,4,2}},
{8,279,1801,{10,12,8,16,14,6,4,2}},
{9,773,5869,{10,2,16,6,20,4,12,14,28}},
{10,1856,15919,{4,14,22,12,2,18,10,6,26,24}},
{11,3724,34883,{14,16,6,20,10,12,2,18,42,4,24}},
{12,6999,70639,{18,6,4,20,22,8,12,24,16,14,10,30}},
{13,7000,70657,{6,4,20,22,8,12,24,16,14,10,30,18,2}},
{14,19205,214867,{16,8,22,26,4,24,20,6,58,12,14,10,36,18}},
{15,19205,214867,{16,8,22,26,4,24,20,6,58,12,14,10,36,18,2}},
{16,184163,2515871,{2,6,18,10,14,30,16,54,36,20,12,34,74,4,8,24}},
{17,280103,3952733,{26,4,8,42,60,36,10,18,30,14,16,12,2,6,34,20,22}},
{18,849876,13010143,{4,12,32,6,16,80,24,22,72,8,28,30,84,2,46,14,10,26}},
{19,1870722,30220163,{8,36,12,54,4,50,16,18,26,10,32,28,14,22,6,2,70,20,42}}

Two more terms from David Newman:
{20,3570761,60155567,{30,14,48,28,56,16,2,46,12,38,4,6,32,24,10,18,8,22,20,36}}
{21,4114341,69931991,{2,10,8,30,18,22,32,16,42,36,20,46,6,44,48,40,26,12,54,4,50}}

More terms:
{22,11271072, 203674907,{30,12,20,24,4,32,34,14,16,6,62,18,22,8,58,42,38,40,2,10,48,26}}
{23,55282774,1092101119,{48,24,30,58,20,4,74,52,12,2,114,22,18,42,44,10,8,36,46,56,90,28,6}}
{24,68256040,1363592621,{56,10,2,84,30,58,8,60,18,12,54,24,52,42,20,96,4,26,46,14,22,32,6,40}}
{25,68256041,1363592677,{10,2,84,30,58,8,60,18,12,54,24,52,42,20,96,4,26,46,14,22,32,6,40,56,16}}
{26,104011359,2124140323,{48,40,152,46,2,28,14,4,72,12,20,58,140,78,30,16,56,42,10,6,62,60,34,36,24,108}}

Difference between neighbor terms not less than d

Difference between neighbor primes not less than d (say,d=100)
a(n) = smallest prime p not in the sequence such that |p-a|>=d; 
case d=100, a(1)=2;

First terms (from 1001 calculated):   
2,103,3,107,5,109,7,113,11,127,13,131,17,137,19,139,23,149,29,151,31,157 
Fixed points:
2,239,409,757,971,1217,1471,1667,1907,1997,2273,2441,2579,2903,3089,
3319,3373,3511,3733,3917,4003,4211,4373,4597,4637,4931,5171,5351,5399,
5431,5657,5857,6079,6269,6551,6977,7193,7457,7741.

First 300 terms of A025415

First 300 terms of A025415 Least sum of 3 distinct nonzero squares in exactly n ways.

{{1,14},{2,62},{3,101},{4,161},{5,206},{6,314},{7,341},{8,446},{9,689},{10,734},{11,854},{12,1106 200 triples {x, y, z}, 0 < x < y < z, such that x^2 + y^2 + z^2 = 176294 = A025415(200)

{2, 27, 419}, {2, 49, 417}, {2, 153, 391}, {2, 211, 363}, {2, 221, 357

Sum and difference of semiprimes are semiprimes.

Least pair is (10, 4): 14 and 6 both sp.

More n's s.t. n +/- are sp: 10, 91, 115, 119, 205, 209, 213, 217, 291, 295, 299, 305, 323, 407, 411, 485, 489, 493, 497, 501, 515, 533, 685, 699, 703, 717, 749, 767, 785, 789, 803, 917, 955, 989, 1007, 1077, 1115, 1137, 1141, 1145, 1195, 1199, 1203, 1207, 1257, 1267, 1333, 1343, 1347, 1351, 1383, 1387, 1389, 1393, 1397, 1401, 1461, 1465, 1469, 1473, 1513, 1565, 1685, 1707, 1731, 1735, 1761, 1765, 1799, 1803, 1839, 1853, 1919, 1923, 1941, 1963, 1981

All are odd (except of 10) as, for even n, one of (n-4, 4, n+4) is divisible by 3 and hence by 6 (and can't be sp).

A220952: Some trivial observations

A220952 Some trivial observation(s):

parities of a(n) and n are the same (hence first differences are odd)

sequence is rearrangement of non-negative integers

for odd squares n's a(n)=2n-1(?)

Look also at (symmetric) graph of abs(a(n)-n):

...nothing fascinating yet :(

A225752: failed submission

Just to keep in history... one of my (many) withdrawn submissions A225752 allocated for Zak Seidov (history; edit; published version)

1. 4 by Zak Seidov at Tue May 14 13:28:01 EDT 2013

NAME a(n) = number of prime digits (2,3,5,7) in (decimal representation of) n^2. allocated for Zak Seidov DATA 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 3, 2, 1, 2, 1, 0, 0, 0, 2, 2, 2, 1, 2, 1, 0, 0, 0, 1, 0, 1, 3, 1, 1, 0, 2, 0, 0, 1, 0, 1, 3, 1, 2, 2, 1, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 3, 1, 1, 0, 3, 2, 0, 1, 1, 0, 1, 1, 3, 2, 3, 3, 2, 0, 1, 0, 1, 2, 0, 2, 4, 2, 2, 2, 2, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 3, 2, 2, 1, 4, 2, 1, 2, 0, 0, 0, 0, 2, 3, 3, 2, 1, 1, 0, 0, 1 OFFSET 1,5 COMMENTS The sequence is unbound. EXAMPLE The first non-zero term is a(5) = 2 because 5^2 = 25 (2 prime digits); a(6) = 1 because 6^2 = 36 (1 prime digit). MATHEMATICA Table[id = IntegerDigits[n^2]; Sum[Count[id, x], {x, {2, 3, 5, 7}}], {n, 150}] CROSSREFS Cf. A191486 Squares using only the prime digits (2, 3, 5, 7). A030485 Squares composed of digits {2, 5, 7}. KEYWORD nonn,changed allocated AUTHOR Zak Seidov, May 14 2013 STATUS proposed approved

1. 3 by Zak Seidov at Tue May 14 08:57:14 EDT 2013

STATUS editing proposed Discussion Tue May 14 13:13 Joerg Arndt: "base" (and, to add some opinion, contrived). 13:43 Charles R Greathouse IV: Can we have a capsule proof that it is unbounded? A naive counting argument won't work, since the set of primes is 'smaller' than the set of numbers composed of nonprime digits.

1. 2 by Zak Seidov at Tue May 14 08:56:19 EDT 2013

NAME allocateda(n) = number of prime fordigits (2,3,5,7) in (decimal Zakrepresentation Seidovof) n^2. DATA 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 3, 2, 1, 2, 1, 0, 0, 0, 2, 2, 2, 1, 2, 1, 0, 0, 0, 1, 0, 1, 3, 1, 1, 0, 2, 0, 0, 1, 0, 1, 3, 1, 2, 2, 1, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 3, 1, 1, 0, 3, 2, 0, 1, 1, 0, 1, 1, 3, 2, 3, 3, 2, 0, 1, 0, 1, 2, 0, 2, 4, 2, 2, 2, 2, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 3, 2, 2, 1, 4, 2, 1, 2, 0, 0, 0, 0, 2, 3, 3, 2, 1, 1, 0, 0, 1 OFFSET 1,5 COMMENTS The sequence is unbound. EXAMPLE The first non-zero term is a(5) = 2 because 5^2 = 25 (2 prime digits); a(6) = 1 because 6^2 = 36 (1 prime digit). MATHEMATICA Table[id = IntegerDigits[n^2]; Sum[Count[id, x], {x, {2, 3, 5, 7}}], {n, 150}] CROSSREFS Cf. A191486 Squares using only the prime digits (2, 3, 5, 7). A030485 Squares composed of digits {2, 5, 7}. KEYWORD allocated nonn AUTHOR Zak Seidov, May 14 2013 STATUS approved editing

How to save large result of search in OEIS?

Sorry, I couldn't put this Q in Q/A page - tricky thing to me :(

If I have many-pages of search results in OEIS (and I need only data format) -

how can I save it in my notebook (as .txt file) to further working off-line? Thanks!

Interesting links

http://www.pbs.org/wgbh/nova/physics/prime-questions.html

https://twitter.com/ZakSeidov

Primes that are the sum of three distinct positive odd squares

All are == 3 mode 4. Cf. A125516 Prime numbers that are the sum of three distinct positive squares. Cf. A242675 Smallest prime with exactly n representations as the sum of three distinct positive squares.

59, 83, 107, 131, 139, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283, 1291, 1307, 1427, 1451, 1459, 1483, 1499, 1523, 1531, 1571, 1579, 1619, 1627, 1667, 1699, 1723, 1747, 1787, 1811, 1867, 1907, 1931, 1979, 1987, 2003, 2011, 2027, 2083, 2099, 2131, 2179, 2203, 2243, 2251, 2267, 2339, 2347, 2371, 2411, 2459, 2467, 2531, 2539, 2579, 2659, 2683, 2699, 2707, 2731, 2803, 2819, 2843, 2851, 2939, 2963, 2971, 3011, 3019, 3067, 3083, 3163, 3187, 3203, 3251, 3259, 3299, 3307, 3323, 3331, 3347, 3371, 3467, 3491, 3499, 3539, 3547, 3571, 3643, 3659, 3691, 3739, 3779, 3803, 3851, 3907, 3923, 3931, 3947, 4003, 4019, 4027, 4051, 4091, 4099, 4139, 4211, 4219, 4243, 4259, 4283, 4339, 4363, 4451, 4483, 4507, 4523, 4547, 4603, 4643, 4651, 4691, 4723, 4787, 4931, 4987, 5003, 5011, 5051, 5059, 5099, 5107, 5147, 5171, 5179, 5227, 5323, 5347, 5387, 5419, 5443, 5483, 5507, 5531, 5563, 5651, 5659, 5683, 5779, 5827, 5843, 5851, 5867, 5923, 5939, 5987, 6011, 6043, 6067, 6091, 6131, 6163, 6203, 6211, 6299, 6323, 6379, 6427, 6451, 6491, 6547, 6563, 6571, 6619, 6659, 6691, 6763, 6779, 6803, 6827, 6883, 6899, 6907, 6947, 6971, 7019, 7027, 7043, 7187, 7211, 7219, 7243, 7283, 7307, 7331, 7411, 7451, 7459, 7499, 7507, 7523, 7547, 7603, 7643, 7691, 7699, 7723, 7867, 7883, 7907, 7963, 8011, 8059, 8123, 8147, 8171, 8179, 8219, 8243, 8291, 8363, 8387, 8419, 8443, 8467, 8539, 8563, 8627, 8699, 8707, 8731, 8747, 8779, 8803, 8819, 8867, 8923, 8963, 8971, 9011, 9043, 9059, 9067, 9091, 9187, 9203, 9227, 9283, 9323, 9371, 9403, 9419, 9467, 9491, 9539, 9547, 9587, 9619, 9643, 9739, 9787, 9803, 9811, 9851, 9859, 9883, 9907, 9923, 9931, 10067, 10091, 10099, 10139, 10163, 10211, 10243, 10259, 10267, 10331, 10427, 10459, 10499, 10531, 10627, 10651, 10667, 10691, 10723, 10739, 10771, 10859, 10867, 10883, 10891, 10939, 10979, 10987, 11003, 11027, 11059, 11083, 11131, 11171

Me in arxiv.org

1	 2004astro.ph..7175S	

07/2004 Seidov, Zakir F. The generalized Roche model

2 2004ApJ...603..283S 03/2004 Seidov, Zakir F. The Roche Problem: Some Analytics

3 2004astro.ph..2130S 02/2004 Seidov, Zakir F. Lane-Emden Equation: perturbation method

4 2004astro.ph..1359S 1.000 01/2004 Seidov, Zakir F. The quasi-incompressible planet: some analytics

5 2003astro.ph.10573S 1.000 10/2003 A X U H Seidov, Zakir F. On the angular momentum of the neutron star 6 2001astro.ph..7395S 1.000 07/2001 A X U H Seidov, Zakir F. Lane-Emden Equation: Picard vs Pade 7 2000astro.ph..3430S 1.000 03/2000 A X R C U H Seidov, Zakir F. Rotation, mass loss and pulsations of the star: an analytical model 8 2000astro.ph..3239S 1.000 03/2000 A X U H Seidov, Zakir F. Gravitational potential energy of simple bodies: the method of negative density 9 2000astro.ph..3233S 1.000 03/2000 A X H Seidov, Zakir F. Gravitational potential energy of simple bodies: the homogeneous bispherical concavo-convex lens 10 2000astro.ph..3064S 1.000 03/2000 A X R C U H Seidov, Zakir F.; Skvirsky, P. I. One relation for self-gravitating bodies 11 2000math......2134S 1.000 02/2000 A X U Seidov, Zakir F. Random triangle problem: geometrical approach 12 2000astro.ph..2496S 1.000 02/2000 A X R C U H Seidov, Zakir F.; Skvirsky, P. I. Gravitational potential and energy of homogeneous rectangular parallelepiped 13 1999astro.ph.12039S 1.000 12/1999 A X C H Seidov, Zakir F. Surprises of phase transition astrophysics 14 1999astro.ph.11489S 1.000 11/1999 A X C U H Seidov, Zakir F. Pulsations and stability of stars with phase transition 15 1999math......7064S 1.000 07/1999 A X U Seidov, Zakir F. Permutations and primes

16 1999astro.ph..7136S 07/1999 Seidov, Zakir F. "Non-1/r" Newtonian gravitation and stellar structure

17 1998astro.ph.10372E 10/1998 Eichler, David; Seidov, Zakir F. Planck Scale Mixing and Neutrino Puzzles

18 1998astro.ph..8078S 08/1998 Seidov, Zakir F. Isotope Separation and Solar Neutrino Experiments

19 1998SoPh..178...29M 00/1998 Mamedov, Sabir G.; Seidov, Zakir F.

p+2 and q+2 are products of n primes

Least prime p such that p+2 and q+2 are products of n primes (q is next prime after p):

3,19,271,1069,15923,24373,892123,2773123,49453123,1272578119

5 and 7 are both primes, 21 and 25 are both semiprimes, 273 and 279 are both 3APs, etc.

A299022

Rare case of collective madness of Editors: see history of my edits:(

Upd Actually it was (not that rare) case of my mind-loss. Mea culpa!!!

My 8 nice sequences, as on Sep 16 2018

https://oeis.org/search?q=keyword%3Anice+author%3ASeidov&sort=&language=english&go=Search

Semiprimes of the form 4+p^2, p prime

365, 533, 965, 1685, 1853, 2813, 5045, 6245, 6893, 11453, 11885, 12773, 17165, 22205, 22805, 24653, 29933, 32765, 36485, 38813, 49733, 51533, 58085, 63005, 66053, 113573, 124613, 143645, 146693

Corresponding values of primes p:

19, 23, 31, 41, 43, 53, 71, 79, 83, 107, 109, 113, 131, 149, 151, 157, 173, 181, 191, 197, 223, 227, 241, 251, 257, 337, 353, 379, 383, 397, 409, 421, 443, 457, 467

Subsequence of A103558. Cf. A001358, A103558, A103686.

Replace one's by two's

Primes which become larger prime under map 1 => 2.

Only primes having one's are considered.

13, 19, 113, 127, 139, 157, 163, 193, 919, 1019, 1039, 1063, 1069, 1087, 1103, 1187, 1193, 1237, 1297, 1399, 1423, 1447, 1459, 1543, 1549, 1579, 1609, 1657, 1663, 1693, 1699, 1753, 1777, 1789, 1879, 1999, 2017, 2137, 2143, 2719, 2917, 3109, 3119, 3313, 3319, 3517, 3613, 3919, 4129, 4153, 4159, 4219, 4513, 5179, 5197, 6163, 6199, 7129, 7219, 7717, 8419, 9103, 9109, 9127, 9157, 9613, 9619, 10133, 10169, 10313, 10333, 10357, 10369, 10399, 10477, 10627, 10639, 10663, 10753, 10789, 10903, 10939, 11003, 11027, 11093, 11119, 11159, 11173, 11177, 11273, 11279, 11369, 11447, 11483, 11549, 11699, 11717, 11777, 11783, 11807

Terms followed by minimal gap 2: 117617, 167117, 171167, 1010717, 1014317, 1117607, 1120517, 1196717, 1247117, 1341017, 1710197, 1891187. Corresponding indices: 633, 1097, 1134, 3692, 3727, 4533, 4571, 5229, 5591, 6283, 8909, 10120

Replace two's by one's

Primes which become lesser prime under map 2 => 1.

Only primes having two's are considered.

23, 29, 223, 239, 251, 257, 263, 281, 293, 821, 929, 1217, 2029, 2039, 2063, 2069, 2087, 2153, 2203, 2251, 2281, 2287, 2293, 2381, 2399, 2447, 2459, 2521, 2531, 2543, 2549, 2579, 2609, 2657, 2663, 2693, 2699, 2741, 2753, 2777, 2789, 2801, 2861, 2879, 2999, 3209, 3229, 3323, 3329, 3527, 3623, 3929, 4211, 4253, 4259, 4523, 5021, 5279, 5297, 6263, 6299, 7727, 8221, 8291, 8429, 9203, 9209, 9257, 9281, 9521, 9623, 9629, 10211, 10259, 11213, 11261, 11273, 11321, 12071, 12113, 12119, 12149, 12161, 12197, 12227, 12277, 12421, 12437, 12491, 12497, 12503, 12689, 12743, 12823, 12941, 12953, 12959, 13259, 13421, 13523, 13721, 14207, 14243, 14249, 14723, 15287, 15299, 15329, 15629, 15923, 16421, 16427, 16529, 17021, 17207, 17291, 17327

Minimal differences 2 occur at

22271, 24419, 201119, 202127.

FB Z85: (q + 2) is multiple of ( p+ 2)

For each odd prime p, q is the smallest prime > p such that (q + 2) is multiple of (p + 2). Table gives first 100 values of p, q and (q+2)/(p+)2. Minimal value of (q+2)/(p+2) is 3 but is it any limit for larger values of (q+2)/(p+2)?

{{3, 13, 3}, {5, 19, 3}, {7, 43, 5}, {11, 37, 3}, {13, 43, 3}, {17, 131, 7}, {19, 61, 3}, {23, 73, 3}, {29, 277, 9}, {31, 97, 3}, {37, 193, 5}, {41, 127, 3}, {43, 223, 5}, {47, 439, 9}, {53, 163, 3}, {59, 181, 3}, {61, 313, 5}, {67, 619, 9}, {71, 509, 7}, {73, 223, 3}, {79, 241, 3}, {83, 593, 7}, {89, 271, 3}, {97, 691, 7}, {101, 307, 3}, {103, 313, 3}, {107, 761, 7}, {109, 331, 3}, {113, 1033, 9}, {127, 643, 5}, {131, 397, 3},{137, 971, 7}, {139, 421, 3}, {149, 3169, 21}, {151, 457, 3}, {157, 1429, 9}, {163, 823, 5}, {167, 1181, 7}, {173, 523, 3}, {179, 541, 3}, {181, 547, 3}, {191, 577, 3}, {193, 1753, 9}, {197, 1789, 9}, {199, 601, 3}, {211, 1063, 5}, {223, 673, 3}, {227, 1601, 7}, {229, 691, 3}, {233, 2113, 9}, {239, 3613, 15}, {241, 727, 3}, {251, 757, 3}, {257, 1811, 7}, {263, 2383, 9}, {269, 811, 3}, {271, 3001, 11}, {277, 1951, 7}, {281, 1979, 7}, {283, 853, 3}, {293, 883, 3}, {307, 1543, 5}, {311, 937, 3}, {313, 2203, 7}, {317, 4783, 15}, {331, 997, 3}, {337, 1693, 5}, {347, 2441, 7}, {349, 1051, 3}, {353, 1063, 3}, {359, 4691, 13}, {367, 3319, 9}, {373, 1123, 3}, {379, 4951, 13}, {383, 1153, 3}, {389, 1171, 3}, {397, 1993, 5},

{401, 2819, 7}, {409, 1231, 3}, 
{419, 5471, 13}, {421, 2113, 5}, 
{431, 1297, 3}, {433, 1303, 3}, 
{439, 1321, 3}, {443, 4003, 9}, 
{449, 4057, 9}, {457, 2293, 5}, 
{461, 9721, 21}, {463, 3253, 7}, 
{467, 4219, 9}, {479, 4327, 9}, 
{487, 7333, 15}, {491, 3449, 7}, 
{499, 2503, 5}, {503, 3533, 7}, 
{509, 1531, 3}, {521, 1567, 3}, 
{523, 3673, 7}, {541, 1627, 3},
{541,1627,3}, {547,6037,11}.

FB Z87

FB Z87 2, 7, 67, 607, 49207, 35872267,

235357947067, 7385781766377084246607,

18773666287449312009866789895643867,

429481267276676109197050941471351060585089601667

Prime numbers in the sequence a(1) = 2 and a(n + 1) = 3*a(n) + 1. All odd terms == 7 mod 10.

Each 3 from 4

Four consecutive semiprimes sum of each 3 of them is a semiprime {9, 10, 14, 15}, {1057, 1059, 1067, 1073}, {2165, 2167, 2171, 2173}, {2827, 2831, 2839, 2841}, {3667, 3669, 3679, 3683}, {5998, 5999, 6001, 6002}, {7694, 7697, 7702, 7706}, {8851, 8857, 8859, 8871}, {8857, 8859, 8871, 8873}, {9071, 9073, 9077, 9079}, {13169, 13173, 13191, 13193}, {14333, 14339, 14349, 14351}, {14494, 14501, 14506, 14507}, {14761, 14765, 14777, 14785}, {14799, 14803, 14809, 14811}, {15303, 15305, 15311, 15321}, {16295, 16297, 16307, 16309}, {17543, 17547, 17553, 17555}

n, n+1 and p=2n+1 are semiprime, semiprime and prime respectively

n: 9, 14, 21, 33, 86, 141, 158, 326, 393, 453, 933

p: 19, 29, 43, 67, 173, 283, 317, 653, 787, 907, 1867

Cf. A40, A1358, A63644, A79153

Numbers n such that n, n+1, n+2 and n+3 are products of 6 primes

8706123, 24463374,32442848,32942943, 36782289,48580623,55486248,57476573, 59600365,59757774,62481222,62664810, 62884590,63262374,63728124,64724373, 65159575,65450824,69362487,70302087, 70370223,70785924,71494773,72060272, 72503682,73256910,73638422,74066874, 74361858,75356070,75414702,78676623, 83349123,84405123,88094895,92349423, 93803982,96192782,97757955,97822374, 98474661,99104550,

Cf. A124729 Numbers n such that n, n+1, n+2 and n+3 are products of 5 primes.

MyA337837

Numbers k such that Omega(3^k - 2) = Omega(3^k + 2) where Omega is A001222.

2, 4, 12, 18, 20, 28, 30, 31, 34, 35, 38, 44, 45, 49, 50, 58, 60, 75, 79, 97, 100, 103, 111, 113, 118, 120, 135, 141, 153, 154, 156, 166, 168, 171, 178, 181, 204, 219, 220, 239, 245, 247, 254, 260, 267, 269, 280, 286, 298, 307, 313

OFFSET1,1

COMMENTS The corresponding values of Omega: 1, 1, 2, 3, 2, 2, 3, 3, 3, 3, 4, 2, 3, 3, 3, 3, 2, 6, 5, 4, 3, 4, 4, 4, 2, 4, 3, 3, 7, 4, 2, 4, 4, 4, 4, 5, 5, 5, 3, 5, 5, 6, 5, 6, 4, 5, 4, 5, 7, 6, 8.

EXAMPLE 2 is a term since Omega(3^2 - 2) = Omega(7) = 1, and Omega(3^2 + 2) = Omega(11) = 1.

MATHEMATICA Select[Range[200], PrimeOmega[3^#-2] == PrimeOmega[3^#+2]&]

(PARI) for (k = 1, 200, if ((m = bigomega (3^k - 2)) == bigomega (3^k + 2), print (k ", " m ", ")))

Cf. A001222, A014224, A051783, A058481, A168607. AUTHOR Zak Seidov, Sep 25 2020 EXTENSIONS a(36)-a(51) from Amiram Eldar, Sep 25 2020

My A337463

Numbers n such that n, n+1, n+2 and n+3 are products of 6 primes (A046306).

8706123, 24463374, 32442848, 32942943, 36782289, 48580623, 55486248, 57476573, 59600365, 59757774, 62481222, 62664810, 62884590, 63262374, 63728124, 64724373, 65159575, 65450824, 69362487, 70302087, 70370223, 70785924, 71494773, 72060272, 72503682, 73256910, 73638422, 74066874, 74361858 OFFSET 1,1 COMMENTS a(1) = 8706123 = A067821(4). EXAMPLE 8706123 = 3^4*19*5657, 8706124 = 2*2*7*7*43*1033, 8706125 = 5^3*17*17*241, 8706126 = 2*3*11*13*73*139. Cf. A067821, A046306. AUTHOR Zak Seidov, Sep 29 2020

My A337219

a(n) is the least positive number k such that 3^n + k is n-almost prime (first n-almost prime after 3^n).

2, 1, 1, 3, 9, 7, 21, 63, 157, 471, 5, 15, 45, 135, 405, 1215, 3645, 10935, 32805, 98415, 295245, 885735, 2657205, 4409119, 2741597, 8224791, 16285765, 15302863, 45908589, 137725767, 77632981, 232898943, 161825917, 485477751, 1456433253, 3027122479, 1565174669

OFFSET 1,1

COMMENTS Often a(n+1)/a(n) = 3.

MATHEMATICA a[n_] := Module[{k = 1}, While[PrimeOmega[3^n + k] != n, k++]; k]; Array[a, 20] (* Amiram Eldar, Sep 18 2020 *)

Cf. A078843.

AUTHOR Zak Seidov, Sep 14 2020

EXTENSIONS a(27)-a(37) from Daniel Suteu, Sep 14 2020

Six consecutive semiprimes starting with k are congruent to {1, 2, 3, 4, 5, 6} (mod 7).

60313, 469582, 897513, 1082362, 1338541, 1781305, 1841001, 1943327, 2278529, 2524957

The least semiprime that is the sum of n consecutive semiprimes.

The least semiprime that is the sum of n consecutive semiprimes. 4, 10, 25, 39, 69, 58, 133, 122, 249, 209, 185, 219, 254, 327, 458, 377, 473, 579, 745, 589, 951, 898, 1047, 843, 917, 1382, 1157, 1243, 1247, 1678, 1514, 1895, 1703, 1707, 2138, 2147, 2599, 2157, 2509, 2515, 2519, 2642, 2771, 3566, 4126, 3317, 3599, 3891, 4198, 3755, 4369, 4223, 4227, 4713, 5403, 5242, 5251, 5078, 5999, 5629, 5633, 6609, 6423, 6429, 6433, 6638, 7666, 7053, 8306, 9167, 9865, 8339, 8345, 8566, 9249, 9974, 9031, 9754, 10749, 10261, 11281, 11038, 11569, 11053, 11857, 14086, 12153, 12442, 13019, 13606, 14483, 13917, 13631, 15709, 14531, 16051, 16679, 15151, 16403, 15469

4, 10 = 4 + 6, 25 = 6 + 9 + 10, 39 = 6 + 9 + 10 + 14.

Odd k: k, k+2 , k+4 and k+6 are products of 4 primes

346575,355923,374011,402741, 420561,422301,532623,542963, 670941,704665,704667,704669, 704973,730245,748161,759939, 773851,776961,823401,885153, 888093,926001,951573,966663

Prime Puzzle 7, 31, 71, 463, 859, 443, 647, 7867, 4003, 7193, 6883, 9103, 3527, 60631, 14947, 31271, 28793, 16349, 85627, 165857, 34603, 39343, 132967, 120863, 176347, 20807, 88169, 23327, 172439, 28703, 99823, 71789, 114259, 248533, 404983, 237071, 307093, 54443, 173347, 309671, 388789, 69143, 73727, 228127, 548843, 503869, 470531, 303727, 103967, 320143, 890993, 460793, 483929, 145799, 450403, 954277, 1160867, 179903, 194687, 793769, 446189, 233879, 726619, 739183, 790519, 565469, 595949, 955183, 362903, 1193443, 410387, 1264063, 1280659, 466547, 490019, 1183709

Pairs {p,q} of consecutive primes such that p^2+p*q+q^2 and p^2+3*p*q+q^2 are prime.

Pairs {p,q} of consecutive primes such that p^2+p*q+q^2 and p^2+3*p*q+q^2 are prime. {2, 3},{5, 7},{11, 13},{79, 83},{229, 233}, {389, 397},{617, 619},{809, 811}, {877, 881},{1193, 1201}, {1447, 1451},{1931, 1933}, {2539, 2543},{2663, 2671}, {2687, 2689},{2741, 2749}, {2969, 2971},{3041, 3049}, {3229, 3251},{3373, 3389}, {3389, 3391},{3463, 3467}, {3613, 3617},{4271, 4273}, {4783, 4787},{5449, 5471}, {5657, 5659},{7129, 7151}, {8167, 8171},{8269, 8273}, {8861, 8863},{9829, 9833}, {11159, 11161},{13217, 13219}, {14423, 14431},{15107, 15121}, {16829, 16831},{17209, 17231}, {17791, 17807},{18097, 18119}, {19541, 19543},{20443, 20477}, {22013, 22027},{23117, 23131}, {23909, 23911},{24179, 24181}, {25073, 25087},{26951, 26953}, {27017, 27031},{28123, 28151}, {29021, 29023},{30703, 30707}, {31223, 31231},{32159, 32173}, {32257, 32261},{33199, 33203}, {33863, 33871},{35897, 35899}, {39979, 39983},{40189, 40193}, {40361, 40387},{40433, 40459}, {41213, 41221},{43117, 43133}, Apparently {p=2,q=3} is the only pair {p,q} such that also p^2 + 5*p*q +q^2 is a prime.

Semiprimes with semiprime digits

{469, 649, 694, 4469, 4694, 4699, 4946, 6499, 6649, 6694, 9446, 9466, 9469, 9946, 44669, 44966, 44969, 46469, 46946, 46969, 46994, 46999, 49466, 49649, 49694, 49699, 49969, 64469, 64649, 64669, 64949, 64994, 66469, 66494, 66694, 69449, 69469, 69494, 69694, 69949, 94469, 94669, 94699, 94969, 96449, 96494, 96649, 96946, 96949, 96994, 99646, 99649, 444694, 446449, 446494, 446669, 446699, 446999, 449669, 449969, 464449, 464469, 464649, 466469, 466499, 466694, 466699, 466999, 469466, 469666, 469669, 469694, 469949, 469999, 494446, 494669, 496469, 496646, 496649, 496699, 496946, 496966, 496969, 499469, 499666, 499699, 499946, 644494, 644649, 644969, 646449, 646469, 646499, 646649, 646946, 646949, 646999, 649466, 649669, 649694, 649699, 649949, 649966, 649994, 649999, 664649, 664946, 664969, 666469, 666499, 666949, 666994, 669449, 669469, 669499, 669946, 669949, 694466, 694469, 694669, 694699, 694949, 694969, 696469, 696494, 696499, 696646, 696649, 944666, 946499, 946699, 946994, 946999, 949469, 949694, 949966, 964466, 964469, 964646, 964669, 964694, 964699, 964949, 964966, 966466, 966494, 966646, 966694, 966949, 969449, 969466, 969469, 969949, 994469, 994646, 994649, 994946, 994969, 996449, 996469, 996499, 996994, 999469, 999946} Semiprimes with semiprime digits: each digits 4, 6, 9 occur at least once. Min[difference[s]]=3.

3-APs: a(n)+a(n+1) is 3-AP

3-APs: a(n)+a(n+1) is 3-AP. 8, 12, 18, 27, 75, 78, 92, 98, 114, 116, 130, 138, 147, 171, 172, 182, 188, 222, 230, 244, 286, 292, 310, 318, 333, 345, 366, 388, 402, 404, 410, 412, 418, 426, 428, 474, 483, 498, 506, 524, 530, 555, 575, 598, 637, 658, 716, 730, 762, 764, 806, 830, 845, 874, 897, 908, 925, 927, 932, 946, 956, 962, 986, 1004, 1005, 1010, 1025, 1030, 1045, 1052, 1058, 1076, 1086, 1102, 1131, 1143, 1146, 1162, 1172, 1173, 1175, 1179, 1194, 1228, 1258, 1265, 1281 8 = 2^3, 12 = 2*2*3, 8+12 = 20 = 2*3*5.

PrimeOmega[{n, n+1}] = {14,13}

761909247, 1173008384, 2213097471, 2415924224,
2575894527, 2757293055, 2847992319, 3029390847,
3107926016, 3228858368, 3528236799, 3754984959,
4263501824, 4533920000, 5931767295, 5992233471,
6448682240, 6521312511

761909247 = 3^11*11*17*23, 761909248 = 2^12*186013.

Nine consecutive primes

Nine consecutive primes such that the total sum is prime and 3 subgroups by 3 terms all have prime sums. {29, 31, 37, 41, 43, 47, 53, 59, 61}, {83, 89, 97, 101, 103, 107, 109, 113, 127}, {389, 397, 401, 409, 419, 421, 431, 433, 439}, {1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213}, {2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351}, {2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591}, {2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749}, {2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819} First case: {29, 31, 37}, {41, 43, 47}, {53, 59, 61} And {29 + 31 + 37 = 97, 41 + 43 + 47 = 131, 53 + 59 + 61 = 173} - all primes.

A342936

A342936 Primes followed by gap 400. 47203303159, 133943178727, 230288509459, 235796599039, 245345683567, 277144433017, 278729935933, 312398312557, 380701947979, 421320335713, 462122822269, 469775998333, 476435929441, 477784206337, 502882883599, 526566124369, 532615065673 All terms are congruent to 1 mod 6. a(1) = 47203303159 = A138198(10) = A123995(15).

20-almost primes as sum of 20 consecutive primes

{36864000, 74649600, 90699264, 99483648, 117571584, 139788288, 156499968, 158269440, 191987712, 278986752}

Each sum is a semiprime

Each sum is a semiprime (product of two primes): 1 + 2 + 3, 4 + 5, 6, 7 + 8, 9, 10 + 11, 12 + 13, 14, 15, 16 + 17, 18 + 19 + 20, 21, 22+23, 24+25, 26, 27+28, 29+30+31+32, 33, 34, 35, 36+37+38, 39, 40+41+42, 43+44, 45+46, 47+48, 49, 50+51+52+53,...,

a(n)-n and a(n+n) are both semiprimes

a(n)-/+n are both semiprimes {5, 8, 7, 10, 9, 15, 28, 14, 13, 16, 15, 21, 22, 20, 19, 22, 21, 28, 58, 26, 25, 36, 32, 33, 40, 32, 31, 34, 33, 39} 4 and 6, 6 and 10, 4-10 are all semiprimes.

(n^2+k^2) is divisible by (n+k+1)

a(n) = the least number k>n such that (n^2+k^2) is divisible by (n+k+1)

3, 10, 21, 36, 55, 10, 105, 20, 171, 210, 41, 300, 59, 406, 21, 92, 595, 118, 741, 820, 163, 990, 41, 1176, 1275, 254, 61, 36, 1711, 1830, 365, 2080, 415, 2346, 2485, 168, 59, 554, 3081, 152, 223, 3570, 713, 188, 55, 126, 4465, 892, 119, 5050, 1009, 96, 175, 402, 6105, 1220, 331, 126, 7021, 7260, 383, 538, 1549, 92, 8515, 78, 633, 1808, 9591, 9870, 337, 10440, 2087, 578, 801, 2264, 215, 346, 12561, 916, 2575, 13530, 2705, 14196, 14535, 118, 15225, 152, 343, 16290, 105, 17020, 175, 242, 397, 648, 18915, 3782, 19701, 20100

a(n) = the least number k>n such that (n^2+k^2) is divisible by (n+k+1)

3, 10, 21, 36, 55, 10, 105, 20, 171, 210, 41, 300, 59, 406, 21, 92, 595, 118, 741, 820, 163, 990, 41, 1176, 1275, 254, 61, 36, 1711, 1830, 365, 2080, 415, 2346, 2485, 168, 59, 554, 3081, 152, 223, 3570, 713, 188, 55, 126, 4465, 892, 119, 5050, 1009, 96, 175, 402, 6105, 1220, 331, 126, 7021, 7260, 383, 538, 1549, 92, 8515, 78, 633, 1808, 9591, 9870, 337, 10440, 2087, 578, 801, 2264, 215, 346, 12561, 916, 2575, 13530, 2705, 14196, 14535, 118, 15225, 152, 343, 16290, 105, 17020, 175, 242, 397, 648, 18915, 3782, 19701, 20100

n(A001358), n+1(A014612)

n is semiprime and n+1 is 3-almost prime (A014612) 26, 49, 51, 62, 65, 69, 74, 77, 91, 115, 123, 129, 146, 169, 185, 187, 194, 206, 221, 235, 237, 254, 265, 267, 274, 278, 289, 291, 309, 321, 355, 362, 365, 386, 398, 403, 411, 417, 422, 427, 437, 451, 454, 469, 473, 482, 493, 497, 505, 517, 529, 533, 538, 554, 573, 581, 589, 597, 614, 626, 662, 667, 669, 681, 721, 723, 746, 753, 758, 763, 771, 781, 785, 789, 794, 813, 843, 866, 889, 893, 901, 905, 914, 926, 934, 955, 961, 985, 993 26 = 2*13, 27 = 3*3*3, 49 = 7*7, 50 = 2*5*5. Cf. A001222, A001358, A014612.

(n+m) | (n*m+1)

a(n) = the least number m>n such that (n*m+1) is divisible by (n+m), or 0, if no such m exists, n=1,2,3,... .

2, 0, 5, 11, 7, 29, 9, 13, 11, 23, 13, 131, 15, 25, 17, 35, 19, 305, 21, 37, 23, 47, 25, 91, 27, 49, 29, 59, 31, 869, 33, 61, 35, 43, 37, 149, 39, 73, 41, 83, 43, 1721, 45, 85, 47, 95, 49, 281, 51, 69, 53, 107, 55, 211, 57, 109, 59, 119, 61, 3539, 63, 121, 65, 131, 67, 269, 69, 133, 71, 143, 73, 5111, 75, 145, 77, 89, 79, 475, 81, 157, 83, 167, 85, 331, 87, 169, 89, 179, 91, 533, 93, 125, 95, 191, 97, 389, 99, 193, 101, 203

Primes mod 17

5 consecutive primes starting with p are congruent to {1,2,3,4,5} mod 17.

28631503, 31896523, 49019671, 73659811, 98496403, 134844511, 201579541, 217034071, 244739413, 248553601, 250555351

Sums of 101, 103 and 105 consecutive primes

Integers that are the sum of 101, 103 and also of 105 consecutive primes. 7326437, 913230375, 1250863941, 1867442601, 2870120343, 5883134315, 6939292055... No primes yet...

Primes congruent to {1,1,1} modulo {11,12,13}

7:53 26/04/2021

3433, 8581, 13729, 20593, 25741, 27457, 29173, 36037, 42901, 44617, 48049, 51481, 53197, 56629, 63493, 72073, 87517, 96097, 97813, 99529, 104677, 108109, 114973, 116689, 120121, 123553, 125269, 145861, 151009, 156157, 159589, 163021, 180181, 192193, 197341, 209353, 216217, 217933, 219649, 224797

Differences are multiples of 11*12*13=1716.

6 consecutive primes == {11,13,17,19,23,29} modulo 30

Primes p such that 6 consecutive primes starting with p are congruent to {11,13,17,19,23,29} modulo 30.

11, 1481, 27701, 165701, 317921, 326141, 397751,

558791, 585911, 661091, 716411, 739391, 959831,

1015361, 1022501, 1068701, 1156031, 1161401,

1246361, 1265861

3 consecutive primes == {1, 2, 3} modulo 31

Primes p such that 3 consecutive primes starting with p are congruent to {1, 2, 3} modulo 31. 21584743, 76751537, 95678773, 132833017, 137077847, 150564521, 179271017

n^m = p2+p3+...pk

k is index of the last prime p and n^m = p2+p3+...pk (sum of odd primes) [k, n^m, p]: [3, 8=2^3, 5] [24, 961=31^2, 89] [28, 1369=37^2, 107] [32, 1849=43^2, 131] [46, 4225=65^2, 199] [296, 263169=513^2, 1949] [5327, 130919364=11442^2, 52081] [12227, 758451600=27540^2, 130729]

primepuzzle?

2, 2, 5, 131, 43, 15619, 281, 6553, 503, 137771, 3061, 244140613, 8179, ...?

Seven primes

11556738791617, 28220722811527, 40311592402447, 42847412581091, 42894320358997, 45460620323231, 46964745279449

4 consecutive primes == {2, 3, 5, 7} (mod 11)

4 consecutive primes starting with primes p are congruent to {2,3,5,7} (mod 11). 82799, 406661, 447779, 490019, 596279, 617971, 654931, 790781, 1286969, 1532291, 1543357, 1775831, 1916939, 1932911, 2220539, 2240977, 2298749, 2307989, 2376629, 2435039, 2458139, 2513579, 2731049, 2775599, 3093851, 3141899, 3213839, 3294337, 3331319, 3351251, 3366497, 3645193, 3689149, 3733259, 3781153, 3981331, 4063589, 4088339, 4187009, 4197239, 4204961, 4924559, 5243911, 5270641, 5316071, 5331581, 5339039, 5396921, 5518339, 5669501, 5768897, 5932709, 6037781, 6441041, 6516149, 6687463, 6696857, 6727481, 6800279, 6853409, 6985871, 7061969, 7227343, 7330633, 7345967, 7463887, 7533209, 7602509, 7864639, 7909739, 8124569, 8279317, 8285257, 8286731, 8310469, 8413649, 8422559, 8468627, 8698153, 8778629, 8789189, 8816711, 9165917, 9221819, 9261089, 9397357, 9434339, 9465601, 9522031, 9553997, 9664679, 9783589, 9871391, 9988519, 9992699

Prime Puzzle

Prime Puzzle

379081, 637001, 1344311, 12043001, 44112311, 1130203021, 201021000001

My A344963 (failed submission)

Integers that can be expressed as the sum of squares of 3 distinct odd primes in exactly 100 ways. 2778699, 4090779, 4209051, 4441059, 4626699, 4910619, 5058291, 5067531, 5070219, 5083659, 5221251, 5270139, 5299371, 5613531, 5659731, 5666619, 5674179, 5852931, 5953899, 5966499, 6008499, 6011859, 6122739, 6158691, 6187419, 6245379, 6317451, 6330819, 6511491, 6544131, 6556539, 6580539, 6695619, 6793731, 6823131, 6838251, 6956691, 6982899, 7037499, 7079331, 7195419, 7261779, 7266819, 7289499, 7367619, 7389459, 7408779, 7598811

All terms are congruent to 3 mod 24.

MATHEMATICA ta = Table[Prime[k]^2 + Prime[k1]^2 + Prime[k2]^2, {k, 1, 320}, {k1, k + 1, 320}, {k2, k1 + 1, 320}]; #1 & /@ Select[Split[Sort[Flatten[ta]]], Length[#] == 100 &]

Zak Seidov, Jun 03 2021

n and n+1 have the same OMEGA (bigomega)

2,9,14,21,25,27,33,34,38,44,57,75,85,86,93,94,98,116,118,121,122,124,133,135, 141,142,145,147,153,158,164,170,171,174,177,201,202,205,213,214,217,218,230, 244,245,253,284,285,296,298,301,302,326,332,334,350,356,361,369,375,381,387, 393,394,425,428,429,434,435,445,446,453,459,474,481,501,506,507,514,526,530, 537,542,548,553,555,565,574,584,595,602,603,604,605,609,620,622,627,633,634, 637,638,645,651,657,694,697,698,706,710,715,717,724,741,745,766,778,793,795, 802,805,817,819,824,833,836,841,842,844,855,865,873,875,878,898,902,908,913, 915,921,922,931,933,944,956,958,962,963,969,973,986

Zak Seidov 26 февраля 2016 г. · Это видят: Доступно всем Z2 Permutation of semiprimes : a (n) + a (n + 1) is a (m). 4, 6, 9, 25, 10, 15, 34, 21, 14, 35, 22, 33, 49, 38, 39, 26, 51, 55, 74, 69, 46, 65, 57, 58, 85, 93, 62, 115, 86, 91, 87, 82, 77, 106, 95, 111, 94, 119, 118, 129, 133, 121, 141, 146, 143, 122, 145, 142, 123, 155, 134, 161, 158, 169, 166, 205, 177, 178, 183, 194, 187, 159, 202, 201, 185, 206, 209, 213, 214, 203, 219, 218, 235, 247, 226, 221, 237, 217, 249, 253, 262, 265, 254, 215, 267, 259, 274, 291, 295, 278, 287, 299, 298, 335, 314, 309, 302, 289, 334, 301, 321 a (1) + a (2) = a(5), or 4+6=10, a (2) + a (3) = a(6), or 6+9=15, a (3) + a (4) = a(7), or 9+25=34, a (4) + a (5) = a(10), or 25+10=35. a (5) + a (6) = a(4), or 10+15=25. Fixed points: {k,a(k)=A001358(k)}: {1,4},{2,6},{3,9},{6,15},{14,38},{15,39},{29,86},{36,111}, {55,166},{57,177},{58,178},{59 183},{61,187},{87,274},{93,298}

Set of triprimes: a(n), a(n+1), a(n+1)-a(n) and a(n+1)+a(n) all are triprimes (A014612).

8, 20, 50, 125, 279, 426, 531, 539, 814, 822, 897, 1002, 1010, 1076, 1146, 1209, 1325, 1353, 1398, 1406, 1516, 1558

Balanced primes with gap = 6

53, 157, 173, 257, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1747, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3637, 3733, 4013, 4457, 4597, 4657, 4993, 5107, 5303, 5387, 5563, 5807, 6073, 6263, 6317, 6367, 6863, 6977, 7523, 7583, 7823, 8117, 8713, 8747, 9397, 9473, 10253, 10607, 10657, 10853, 11497, 11807, 11903, 11933, 12497, 12547, 12583, 12647, 12973, 13043, 13177, 13457, 14543, 14747, 15193, 15313, 15467, 15767, 15797, 15907, 16097, 16223, 16427, 16487, 16567, 16937, 16987, 17047, 17327, 17477, 18217, 18433, 19463, 19477, 19577, 19603, 20107, 20123, 20347, 21163, 21997, 22073, 22447, 23327, 23767 47,53,59 are triple of consecutive primes with gap 6: 53-47=59-53=6.

4, 5, 7, 7, 8, 11, 10, 11, 13, 13, ... ?

4, 5, 7, 7, 8, 11, 10, 11, 13, 13, 15, 17, 16, 17, 19, 21, 20, 23, 22, 23, 25, 25, 30, 29, 28, 32, 31, 31, 32, 37, 36, 35, 38, 37, 38, 43, 42, 41, 43, 43, 45, 47, 46, 50, 49, 51, 50, 53, 52, 53, 55, 55, 56, 59, 58, 62, 61, 64, 63, 67, 66, 65, 68, 67, 69, 71, 70, 71, 73, 76, 75, 77, 76, 77, 82, 81, 80, 85, 84, 83, 85, 85, 90, 89, 88, 92, 91, 91, 92, 98, 99, 99, 98, 97, 98, 101, 100, 101, 103, 106, ... ?

Prime Puzzle

5, 19, 101, 197, 727, 677, 3469, 10831, 12697, 10093, 15377, 5477, 16811, 22189, 8837, 78653, 20887, 171167, 17957, 50411, ... ?

Prime Puzzle

5, 19, 101, 197, 727, 677, 3469, 10831, 12697, 10093, 15377, 5477, 16811, 22189, 8837, 78653, 20887, 171167, 17957, 50411, ..., ?

3, 7, 9, 24, 17, 35, 25, 51, 90, 37, . . . , ?

3, 7, 9, 24, 17, 35, 25, 51, 90, 37, ...,?

Primes that are sum of 1000 consecutive semiprimes.

Z221 Primes that are sum of 1000 consecutive semiprimes. 1814161, 1828627, 1839493, 1843111, 1857617, 1893949, 1912139, 1937659, 2139107, 2172283, 2264593, 2305271, 2308967, 2331151, 2368309, 2424173, 2501987, 2557691, 2583709, 2684347, 2717881, 2788783, 2871019, 2874749, 2911973, 2945497, 2949217, 2997503, 3053279, 3057011, 3083233, 3260473, 3286991, 3374089, 3389219, 3543571, 3566177, 3641653, 3679567, 3683357, 3911191, 3926471, 4037729, 4240693, 4275127, 4313329, 4401653, 4432627, 4467377, 4498283, 4721687, 4729331, 4802257, 5066161, 5109283, 5261693

DD powers of 2

Distinct-digits powers of 2 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 1048576, 536870912 fini full

Primes modulo 7

29 2 3 11 5 13 43 23 17 53 19 41

Least new prime number such that first digit of a(n) = last digit of a(n-1)

2,3,31,11,13,37,7,71,17,73,307,79,97,701,101,107,709,907,

Four sums with the same OMEGA

ss = {}; p = 2; q = 3; r = 5;

Do[p = q; q = r; r = NextPrime[r]; If [(po = PrimeOmega[p + q]) == PrimeOmega[p + r] == PrimeOmega[q + r] == PrimeOmega[p + q + r], AppendTo[ss, {p, po}]], {2000}]; ss

{{1559, 3}, {4073, 4}, {5237, 3}, {5987, 3}, {12119, 3}, {14633, 4}}

Z249 Consecutive 3 - APs (a < b)  : b + a and b - a are 3 - APs

{130, 138}, {195, 207}, {222, 230}, {292, 310}, {498, 506}, {582, 590},{670, 678}, {814, 822}, {970, 978}, {1362, 1370}, {1398, 1406}, {1534, 1542}, {1645, 1653}, {1813, 1825}, {1834, 1842}, {1978, 1986}, {2514, 2522}, {2717, 2725}, {2853, 2865}, {2865, 2873}, {2994, 3002}, {3092, 3110}, {3130, 3138}, {3157, 3165}, {3211, 3219}, {3462, 3470}, {3897, 3905}, {4527, 4539}, {4615, 4623}, {4707, 4715}, {4782, 4790}, {5529, 5537}, {6070, 6078}, {6610, 6618}, {7270, 7278}, {7399, 7407}, {7414, 7422}, {7527, 7535}, {7767, 7775}, {8029, 8041}, {8305, 8313}, {8687, 8695}, {8911, 8919} Select[Partition[Select[Range[8, 10000], 3 == PrimeOmega[#] &], 2, 1], 3 == PrimeOmega[#1 + #2] == PrimeOmega[#2 - #1] &] In most cases, b - a = 8(?)

12, 48, 66, 78, 84

12, 48, 66, 78, 84 k^2 - 25 and k^2 + 25 are semiprimes All k's are multiples of 6 (?)

s(n)+s(n-1) is the product of n prime factors (counting with multiplicity )

{2, 3, 7, 11, 13, 19, 197, 251, 389, 1531, 2053, 8699, 12037, 30971, 51973, 132347, 199429, 292091, 363269, 2389243, 3115781, 3962107, 18057989, 48002299, 115575557, 178025723, 199461637, 303854843, 702778117, 1109161211} offset is zero 2+3=5 (prime), 3+7=10=2*5 (semiprime), 7+11=18=2*2*3 (three prime)

LFS of semiprimes

LFS of semiprimes {4, 9, 10, 21, 22, 15, 14, 33, 26, 35, 6, 25, 34, 39} a(n) + a(n + 1) is prime {13, 19, 31, 43, 37, 29, 47, 59, 61, 41, 59 73}

Semiprimes such that a(n) -+ (n-1) are also semiprimes.

4, 10, 25, 94, 115, 206, 221, 298, 391, 478, 511, 526, 551, 586, 655, 694, 703, 758, 779, 934, 949, 974, 989, 993, 1126, 1159, 1418, 1513, 1522, 1555, 1594, 1603, 1658, 1679, 1718, 1769, 2018, 2051, 2066, 2105, 2174, 2195, 2234, 2319, 2462, 2501, 2578, 2587, 2846, 2867, 2906, 2931, 2986, 3007, 3226, 3241, 3274

Four Sums

{1559, 1567, 1571, 3}, {4073, 4079, 4091, 4}, {5237, 5261, 5273, 3}, {5987, 6007, 6011, 3}

Consecutive primes {p,q} such that p, q, q-p, p*q and p+q all have distinct digits

Consider pair of consecutive primes {p,q} such that p, q, q-p, p*q and p+q all have distinct digits. Sequence gives lesser primes p. {2,3,5,19,41,61,137,173,193,241,251,257,281,601,683,2143,2543,2801,2851,3697,8623} This sequence has 21 terms the last being 8623. Subsequence of A356196 and A029743.

Three consecutive semiprimes with distinct digits

s = {85, 93, 213, 217, 841, 1345, 1893, 2305, 2517}

Numbers m such that m, m+1 and m+2 are semiprimes with distinct digits (A320969): 

{85=5*17, 86=2*43, 87=3*29}, {93=3*31, 94=2*47, 95=5*19}, etc.

27, 18, 63, 8, 45

{27, 18, 63, 8, 45, 12, 20, 130, 138, 154, 52, 30, 561, 78, 1194, 1930, 4277, 292, 5343, 26353, 4255, 7847, 34773, 18628}

{5, 3, 13, 1, 10, 2, 4, 31, 32, 36, 12, 7, 136, 19, 302, 486, 1094, 73, 1366, 6763, 1092, 2006, 8924, 4785}

135, 54, 81, 36

{135, 54, 81, 36, 184, 126, 189, 16, 297, 90, 825, 24, 315, 40, 444, 260, 424, 276, 1892, 884, 60, 104, 3627, 390, 10940, 1122, 4338, 156, 8559, 15492, 9177, 21324, 80577, 16276, 585, 259230, 521925, 107430, 52706, 286178, 144627, 409257, 196670, 609054, 753666, 628698}

Dear NJAS, May I edit my User Page? Please allow me to do it! 06:00 Dec 2022

Prime Squares in AP.

Sum of digits of k and sum of digits of k^2 are both a square

0, 1, 9, 10, 13, 18, 22, 31, 45, 90, 100, 103, 112, 121, 130, 180, 202, 211, 220, 301, 310, 351, 423, 450, 900, 1000, 1003, 1012, 1021, 1030, 1044, 1102, 1111, 1120, 1134, 1201, 1210, 1224, 1300, 1314, 1341, 1800, 2002, 2011, 2020, 2043, 2101, 2110, 2142, 2167, 2200, 2214, 2223, 2232, 2322, 2383, 2403, 2412, 2437, 2563, 2617, 2626, 2824, 3001, 3010, 3024, 3042, 3100, 3123, 3132, 3157, 3303, 3312, 3510, 4023, 4041, 4104, 4183, 4203, 4212, 4219, 4221, 4228, 4230, 4311, 4401, 4417, 4426, 4471, 4500, 4687, 4867, 5083, 5137, 5164, 5173, 5263, 5272, 5281, 5974, 5983, 6163, 6244, 6307, 6316, 6667, 6676, 6766, 6883, 6892, 6928, 6964, 7027, 7054, 7126, 7333, 7387, 7414, 7567, 7576, 7594, 7657, 7666, 7738, 8062, 8116, 8233, 8287, 8359, 8413, 8593, 8683, 8728, 8764, 8773, 8809, 8827, 8836, 8863, 8917, 8926, 8944, 9000, 9043, 9268, 9286, 9313, 9367, 9376, 9412, 9457, 9475, 9484, 9583, 9628, 9637, 9664, 9673, 9772, 9781, 9808, 9817, 9907, 9934, 9943, 9999, 10000

Triples of primes {p,q,r}

{3, 5, 17}, {5, 7, 37}, {13, 17, 229}, {43, 47, 2029}, {113, 127, 14449}, {149, 151, 22501}, {157, 163, 25609}, {167, 173, 28909}, {179, 181, 32401}, {199, 211, 42061}, {239, 241, 57601}, {269, 271, 72901}, {337, 347, 116989}, {419, 421, 176401}, {421, 431, 181501}, {509, 521, 265261}, {547, 557, 304729}, {569, 571, 324901}, {613, 617, 378229}, {677, 683, 462409}, {701, 709, 497041}, {829, 839, 695581}, {887, 907, 804709}, {1039, 1049, 1089961}, {1129, 1151, 1299721}, {1187, 1193, 1416109}

p = 3; q = 5; Do[If[PrimeQ[r = (p^2 + q^2)/2], Print[{p, q, r}]]; p = q; q = NextPrime[q], {200}]

Set of primes {p,q,r,P,Q,R,T}

{p = 3; q = 5; r = 7;
for (k = 1, 10^6,
  if (isprime (P = p + q + r) && isprime (Q = p^2 + q^2 + r^2) &&
     isprime (R = p^3 + q^3 + r^3) && isprime (T = p^4 + q^4 + r^4),
    print ( r ", " P ", " Q "," R ", " T ", "));p=q;q=r;
  r = nextprime (r + 1))}

44483, 44491, 1978737323,88020170826739, 3915401258879070227, 84053, 84061, 7064906843,593826612017029, 49912908219854563187, 180233, 180241, 32483934323,5854676928709489, 1055205986892069936227, 312023, 312031, 97358352563,30378045231156319, 9478648807161040696547, 1002143, 1002151, 1004290592483,1006442787188618359, 1008599594081563415818307, 1019723, 1019731, 1039834996763,1060343662369486219, 1081256820422399440700147, 1759553, 1759561, 3096026759843,5447623173302205529, 9585381697453415377717187, 1846913, 1846921, 3411087629603,6299982087190170649, 11635518816598659363126467, 2146043, 2146051, 4605500557883,9883602233667941659, 21210635388347450195507507, 3734183, 3734191, 13944122677523,52069905852193906639, 194438557244862998307345827, 4638533, 4638541, 21515988392123,99802622184321765589, 462937756488508591597784627, 4761143, 4761151, 22668482666483,107927887567984991359, 513860106399098764990270307, 7200203, 7200211, 51842923241243,373279571450122765579, 2687688690193888285995782387, 7444643, 7444651, 55422709397483,412602285556752915859, 3071676716954081696647708307

Prime Puzzle

Fini full sequence 3, 5, 11, 41, 47, 53, 71, 83, 113, 137, 167, 197, 227, 251, 263, 281, 347, 383, 431, 461, 467, 503, 521, 557, 563, 593, 641, 677, 743, 761, 773, 797, 827, 857, 887, 911, 941, 953, 971, 977, 983, 1013, 1019, 1021

On A088730

a(n) is the largest prime factor of A088730(n)

3,13,4733,1806113,1803647,2699538733,

109912203092239643840221,1920647391913,549334763

Six consecutive primes

Six consecutive primes starting at p are {1,2,4,5,7,8} (mod 9) in this order.

56197, 342037, 464941, 534637, 637327, 651169, 698239, 774919, 823789, 1142083

Anyone can submit it to OEIS?

Thx a lot,

Zak

zakseidov@yahoo.com

A071704

Positions and values of local minima in A071704. {{14, 40}, {21, 82}, {27, 136}, {44, 334}, {51, 441}, {53, 486}, {55, 508}, {61, 592}, {65, 688}, {71, 823}, {76, 935}, {84, 1129}, {86, 1165}, {91, 1255}, {96, 1415}, {99, 1500}, {106, 1758}, {111, 1887}, {117, 2071}, {123, 2223}, {127, 2437}, {132, 2615}, {134, 2674}, {137, 2730}, {143, 3049}, {148, 3218}, {150, 3258}, {153, 3365}, {161, 3798}, {165, 4003}, {168, 4096}

"Advise" vs “advice"

Example:

I advise you to take my advice.

Summary:

1. “Advise” is used as a verb and “advice” is used as a noun. 2. The word “Advise” came first than the word “advice”.

Read more: Difference Between Advise and Advice | Difference Between http://www.differencebetween.net/language/difference-between-advise-and-advice/#ixzz7pDEGhVTb