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Search: seq:2,6,8,12,16
Displaying 1-10 of 11 results found. page 1 2
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A008407 Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations. +30
29
0, 2, 6, 8, 12, 16, 20, 26, 30, 32, 36, 42, 48, 50, 56, 60, 66, 70, 76, 80, 84, 90, 94, 100, 110, 114, 120, 126, 130, 136, 140, 146, 152, 156, 158, 162, 168, 176, 182, 186, 188, 196, 200, 210, 212, 216, 226, 236, 240, 246, 252, 254, 264, 270, 272, 278 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Tony Forbes defines a prime k-tuplet (distinguished from a prime k-tuple) to be a maximally possible dense cluster of primes (a prime constellation) which will necessarily involve consecutive primes whereas a prime k-tuple is a prime cluster which may not necessarily be of maximum possible density (in which case the primes are not necessarily consecutive.)
a(n) >> n log log n; in particular, for any eps > 0, there is an N such that a(n) > (e^gamma - eps) n log log n for all n > N. Probably N can be chosen as 1; the actual rate of growth is larger. Can a larger growth rate be established? Perhaps a(n) ~ n log n. - Charles R Greathouse IV, Apr 19 2012
Conjecture: (i) The sequence a(n)^(1/n) (n=3,4,...) is strictly decreasing (to the limit 1). (ii) We have 0 < a(n)/n - H_n < (gamma + 2)/(log n) for all n > 4, where H_n denotes the harmonic number 1+1/2+1/3+...+1/n, and gamma refers to the Euler constant 0.5772... [The second inequality has been verified for n = 5, 6, ..., 5000.] - Zhi-Wei Sun, Jun 28 2013.
Conjecture: For any integer n > 2, there is 1 < k < n such that 2*n - a(k)- 1 and 2*n - a(k) + 1 are twin primes. Also, every n = 3, 4, ... can be written as p + a(k)/2 with p a prime and k an integer greater than one. - Zhi-Wei Sun, Jun 29-30 2013.
The number of configurations that realize this minimal diameter, is A083409(n). - Jeppe Stig Nielsen, Jul 26 2018
REFERENCES
R. K. Guy, "Unsolved Problems in Number Theory", lists a number of relevant papers in Section A8.
John Leech, "Groups of primes having maximum density", Math. Tables Aids to Comput., 12 (1958) 144-145.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..672 (from Engelsma's data)
Thomas J. Engelsma, Permissible Patterns
Tony Forbes and Norman Luhn, k-tuplets
Daniel A. Goldston, Apoorva Panidapu, and Jordan Schettler, Explicit Calculations for Sono's Multidimensional Sieve of E2-Numbers, arXiv:2208.13931 [math.NT], 2022. See H(n) in Table 1 p. 2.
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923. See final section.
T. Tao, Bounded gaps between primes, PolyMath Wiki Project, 2013.
Eric Weisstein's World of Mathematics, Prime Constellation.
FORMULA
s(k), k >= 2, is smallest s such that there exist B = {b_1, b_2, ..., b_k} with s = b_k - b_1 and such that for all primes p <= k, not all residues modulo p are represented by B.
CROSSREFS
Equals A020497 - 1.
Cf. A083409.
KEYWORD
nonn,nice,changed
AUTHOR
T. Forbes (anthony.d.forbes(AT)googlemail.com)
EXTENSIONS
Correction from Pat Weidhaas (weidhaas(AT)wotan.llnl.gov), Jun 15 1997
Edited by Daniel Forgues, Aug 13 2009
a(1)=0 prepended by Max Alekseyev, Aug 14 2015
STATUS
approved
A111051 Numbers m such that 3*m^2 + 1 is prime. +30
7
2, 6, 8, 12, 16, 20, 22, 26, 34, 36, 40, 58, 64, 68, 78, 82, 84, 86, 98, 112, 120, 126, 142, 146, 148, 152, 156, 160, 168, 188, 190, 194, 196, 208, 216, 218, 222, 238, 240, 244, 246, 254, 264, 272, 282, 286, 294, 300, 302, 306, 308, 316, 320, 330, 338, 344, 348 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The resulting primes are the generalized cuban primes of the form (x^3-y^3)/(x-y), x=y+2 (see A002648). - Jani Melik, Jul 18 2007
LINKS
FORMULA
a(n) = sqrt((A002648(n)-1)/3). - Zak Seidov, Feb 04 2016
EXAMPLE
1 + 3*2^2 = 13 = A002648(1) is the 1st prime of this form, so a(1) = 2.
1 + 3*6^2 = 109 = A002648(2) is the 2nd prime of this form, so a(2) = 6.
1 + 3*8^2 = 193 = A002648(3) is the 3rd prime of this form, so a(3) = 8.
If m=98 then 3*m^2 + 1 = 28813 = A002648(19) is prime (the 19th of this form), so 98 is a term (the 19th).
MAPLE
ts_kubpra_ind:=proc(n) local i, tren, ans; ans:=[ ]: for i from 0 to n do tren:=1+3*i^2: if (isprime(tren)='true') then ans:=[ op(ans), i ] fi od: RETURN(ans); end: ts_kubpra_ind(2000); # Jani Melik, Jul 18 2007
MATHEMATICA
Select[Range[400], PrimeQ[3#^2+1]&] (* Harvey P. Dale, Jul 17 2016 *)
PROG
(PARI) is(n)=isprime(3*n^2+1) \\ Charles R Greathouse IV, Feb 07 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Parthasarathy Nambi, Oct 06 2005
EXTENSIONS
More terms from Jani Melik, Jul 18 2007
Edited by N. J. A. Sloane, Sep 28 2007
STATUS
approved
A083769 a(1)=2; for n >= 2, a(n) = smallest even number such that a(1)*a(2)*...*a(n) + 1 is prime. +30
5
2, 6, 8, 12, 16, 10, 4, 30, 26, 22, 24, 14, 50, 42, 18, 64, 46, 60, 32, 36, 20, 34, 28, 108, 48, 44, 68, 282, 90, 54, 76, 62, 180, 66, 132, 86, 74, 38, 58, 106, 120, 52, 244, 94, 100, 82, 138, 156, 98, 72, 172, 150, 248, 154, 166, 114, 162, 126, 124, 208, 222, 324, 212 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Is this a permutation of the even numbers?
For any even positive integers a_1, a_2, ..., a_n, there are infinitely many even positive integers t such that a_1 a_2 ... a_n t + 1 is prime: this follows from Dirichlet's theorem on primes in arithmetic progressions. As far as I know there is no guarantee that the sequence defined here leads to a permutation of the even numbers, i.e. there might be some even integer that never appears in the sequence. However, if the partial products a_1 ... a_n grow like 2^n n!, heuristically the probability of a_1 ... a_n t + 1 being prime is on the order of 1/log(a_1 ... a_n) ~ 1/(n log n), and since sum_n 1/(n log n) diverges we might expect that there should be infinitely many n for which some a_1 ... a_n t + 1 is prime, and thus every even integer should occur. - Robert Israel, Dec 20 2012
LINKS
EXAMPLE
2+1=3, 2*6+1=13, 2*6*8+1=97, 2*6*8*12+1=1153, etc. are primes.
After 200 terms the prime is
224198929826405912196464851358435330956778558123234657623126\
069546460095464785674042966210907411841359152393200850271694\
899718487202330385432243578646330245831108247815285116235792\
875886417750289946171599027675234787802312202111702704952223\
563058999855839876391430601719636148884060097930252529666254\
756431522481046758186320659298713737639441014068272279177710\
551232067814381240340990584869121776471244800000000000000000\
00000000000000000000000000000 (449 digits). - Robert Israel, Dec 21 2012
MAPLE
N := 200: # number of terms desired
P := 2:
a[1] := 2:
C := {seq(2*j, j = 2 .. 10)}:
Cmax := 20:
for n from 2 to N do
for t in C do
if isprime(t*P+1) then
a[n]:= t;
P:= t*P;
C:= C minus {t};
break;
end if;
end do;
while not assigned(a[n]) do
t0:= Cmax+2;
Cmax:= 2*Cmax;
C:= C union {seq(j, j=t0 .. Cmax, 2)};
for t from t0 to Cmax by 2 do
if isprime(t*P+1) then
a[n]:= t;
P:= t*P;
C:= C minus {t};
break;
end if
end do;
end do;
end do;
[seq(a[n], n=1..N)];
MATHEMATICA
f[s_List] := Block[{k = 2, p = Times @@ s}, While[ MemberQ[s, k] || !PrimeQ[k*p + 1], k += 2]; Append[s, k]]; Nest[f, {2}, 62] (* Robert G. Wilson v, Dec 24 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003
EXTENSIONS
More terms from David Wasserman, Nov 23 2004
Edited by N. J. A. Sloane, Dec 20 2012
Comment edited, Maple code and additional terms by Robert Israel, Dec 20 2012
STATUS
approved
A084724 Beginning with 2, the smallest even number greater than the previous term such that every partial product + 1 is a prime. +30
5
2, 6, 8, 12, 16, 18, 20, 22, 26, 36, 42, 44, 100, 120, 124, 162, 168, 174, 192, 218, 272, 278, 338, 364, 380, 392, 502, 512, 532, 560, 594, 614, 698, 790, 814, 838, 922, 938, 1072, 1082, 1092, 1102, 1146, 1182, 1256, 1354, 1360, 1484, 1508, 1566, 1662, 1690 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
MATHEMATICA
ppp[{p_, a_}]:=Module[{n=a+2}, While[!PrimeQ[p*n+1], n=n+2]; {p*n, n}]; NestList[ ppp, {2, 2}, 60][[All, 2]] (* Harvey P. Dale, Aug 12 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 13 2003
EXTENSIONS
More terms from David Wasserman, Jan 03 2005
STATUS
approved
A136513 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of diameter n. +30
5
0, 0, 2, 2, 6, 8, 12, 16, 26, 30, 38, 44, 56, 60, 74, 82, 96, 108, 128, 138, 154, 166, 188, 196, 220, 238, 262, 278, 304, 324, 344, 366, 398, 416, 452, 468, 506, 526, 562, 588, 616, 644, 686, 714, 754, 780, 824, 848, 894, 930, 976, 1008, 1056, 1090, 1134, 1170 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
FORMULA
Lim_{n -> oo} a(n)/(n^2) -> Pi/8.
a(n) = 2 * Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).
a(n) = 2 * A136483(n).
a(n) = (1/2) * A136485(n).
a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (2 * (1 - x)). - Ilya Gutkovskiy, Nov 24 2021
EXAMPLE
a(3) = 2 because a circle centered at the origin and of radius 3/2 encloses (-1,1) and (1,1) in the upper half plane.
MATHEMATICA
Table[2*Sum[Floor[Sqrt[(n/2)^2 -k^2]], {k, Floor[n/2]}], {n, 100}]
PROG
(Magma)
A136513:= func< n | n eq 1 select 0 else 2*(&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
[A136513(n): n in [1..100]]; // G. C. Greubel, Jul 27 2023
(SageMath)
def A136513(n): return 2*sum(isqrt((n/2)^2-k^2) for k in range(1, (n//2)+1))
[A136513(n) for n in range(1, 101)] # G. C. Greubel, Jul 27 2023
(PARI) a(n) = 2*sum(k=1, n\2, sqrtint((n/2)^2-k^2)); \\ Michel Marcus, Jul 27 2023
CROSSREFS
Alternating merge of A136514 and A136515.
KEYWORD
easy,nonn
AUTHOR
Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008
STATUS
approved
A283233 2*A000201. +30
4
2, 6, 8, 12, 16, 18, 22, 24, 28, 32, 34, 38, 42, 44, 48, 50, 54, 58, 60, 64, 66, 70, 74, 76, 80, 84, 86, 90, 92, 96, 100, 102, 106, 110, 112, 116, 118, 122, 126, 128, 132, 134, 138, 142, 144, 148, 152, 154, 158, 160, 164, 168, 170, 174, 176, 180, 184, 186 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where [ ]=floor.
Taking r=1, s=(-1+sqrt(5))/2, t=(1+sqrt(5))/2 gives a=A283233, b=A283234, c=A005843.
LINKS
FORMULA
a(n) = 2*floor(n*r), where r = (1+sqrt(5))/2.
MATHEMATICA
r = 1; s = (-1 + 5^(1/2))/2; t = (1 + 5^(1/2))/2;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}] (* A283233 *)
Table[b[n], {n, 1, 120}] (* A283234 *)
Table[c[n], {n, 1, 120}] (* A005408 *)
PROG
(Python)
from math import isqrt
def A283233(n): return (n+isqrt(5*n**2))&-2 # Chai Wah Wu, Aug 10 2022
CROSSREFS
Cf. A000201, A283234, A005843 (sequential union of A283233 and A283234), A005408.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 03 2017
STATUS
approved
A285342 Positions of 0 in A285341; complement of A285343. +30
4
2, 6, 8, 12, 16, 20, 22, 26, 28, 32, 36, 40, 42, 46, 50, 54, 56, 60, 64, 68, 70, 74, 76, 80, 84, 88, 90, 94, 96, 100, 104, 108, 110, 114, 118, 122, 124, 128, 132, 136, 138, 142, 144, 148, 152, 156, 158, 162, 166, 170, 172, 176, 180, 184, 186, 190, 192, 196 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Conjecture: 0 < n*r - a(n) < 4 for n>=1, where r = 2 + sqrt(2).
LINKS
EXAMPLE
As a word, A285341 = 1011101..., in which 0 is in positions 2,6,8,12,16,...
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 1, 1}}] &, {0}, 10]; (* A285341 *)
u = Flatten[Position[s, 0]]; (* A285342 *)
Flatten[Position[s, 1]]; (* A285343 *)
u/2 (* A285344)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 25 2017
STATUS
approved
A189400 n+[nr/s]+[nt/s]; r=1, s=sqrt(e), t=e. +30
3
2, 6, 8, 12, 16, 18, 22, 25, 28, 32, 35, 38, 41, 45, 48, 51, 55, 57, 61, 64, 67, 71, 73, 77, 81, 83, 87, 90, 93, 97, 100, 103, 107, 110, 113, 116, 120, 123, 126, 129, 132, 136, 139, 142, 146, 148, 152, 156, 158, 162, 165, 168, 172, 175, 178, 181, 184, 188, 191, 194, 197, 201, 204, 207, 211, 214, 217, 221, 223, 227, 231, 233, 237, 240, 243, 247, 249, 253, 256, 259, 263, 266, 269 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
(See A189399.)
LINKS
MATHEMATICA
(See A189399.)
Table[n+Floor[n/Sqrt[E]]+Floor[n*Sqrt[E]], {n, 90}] (* Harvey P. Dale, Aug 20 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 21 2011
STATUS
approved
A057656 Number of points (x,y) in square lattice with (x-1/2)^2+y^2 <= n. +30
2
2, 6, 8, 12, 16, 16, 22, 26, 26, 30, 34, 38, 40, 44, 44, 48, 56, 56, 60, 60, 62, 70, 74, 74, 78, 82, 82, 86, 90, 94, 96, 104, 104, 104, 108, 108, 116, 120, 124, 128, 128, 128, 134, 138, 138, 142, 150, 150, 154, 158, 158, 166, 166, 166, 166, 174 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
LINKS
CROSSREFS
Partial sums of A004020. Cf. A057656, A057961, A057962.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 15 2000
STATUS
approved
A077561 Indices of terms of A025487 which divide the terms. Numbers k such that A025487(k) is a multiple of k. +30
2
1, 2, 6, 8, 12, 16, 20, 24, 32, 45, 48, 66, 84, 90, 96, 108, 120, 140, 144, 150, 154, 162, 168, 175, 180, 192, 198, 200, 216, 220, 224, 240, 252, 264, 280, 288, 300, 315, 324, 336, 360, 375, 390, 396, 432, 486, 504, 525, 570, 576, 594 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3762 from Charlie Neder)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Nov 10 2002
EXTENSIONS
More terms from Ray Chandler, Aug 24 2003
STATUS
approved
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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)