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Search: "n distinct prime factors"
Displaying 1-10 of 133 results found. page 1 2 3 4 5 6 7 8 9 10 ... 14
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A180278 Smallest nonnegative integer k such that k^2 + 1 has exactly n distinct prime factors. +20
15
0, 1, 3, 13, 47, 447, 2163, 24263, 241727, 2923783, 16485763, 169053487, 4535472963, 36316463227, 879728844873, 4476534430363, 119919330795347, 1374445897718223, 106298577886531087 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(12) <= 5755933903. - Donovan Johnson, Aug 27 2012
LINKS
Table of n, a(n) for n=0..18.
FORMULA
a(n) >= sqrt(A185952(n)-1). - Charles R Greathouse IV, Feb 17 2015
a(n) <= A164511(n). - Daniel Suteu, Feb 20 2023
EXAMPLE
a(2) = 3 because the 2 distinct prime factors of 3^2 + 1 are {2, 5};
a(10) = 16485763 because the 10 distinct prime factors of 16485763^2 + 1 are {2, 5, 13, 17, 29, 37, 41, 73, 149, 257}.
MATHEMATICA
a[n_] := a[n] = Module[{k = 1}, If[n == 0, Return[0]]; Monitor[While[PrimeNu[k^2 + 1] != n, k++]; k, {n, k}]]; Table[a[n], {n, 0, 8}] (* Robert P. P. McKone, Sep 13 2023 *)
PROG
(Python)
from itertools import count
from sympy import factorint
def A180278(n):
return next(k for k in count() if len(factorint(k**2+1)) == n) # Pontus von Brömssen, Sep 12 2023
(PARI) a(n)=for(k=0, oo, if(omega(k^2+1) == n, return(k))) \\ Andrew Howroyd, Sep 12 2023
CROSSREFS
Cf. A001221, A002144, A002522, A128428, A164511, A185952, A219017, A219108.
KEYWORD
nonn,hard,more
AUTHOR
Michel Lagneau, Jan 17 2011
EXTENSIONS
a(9), a(10) and example corrected; a(11) added, Donovan Johnson, Aug 27 2012
a(12) from Giovanni Resta, May 10 2017
a(13)-a(17) from Daniel Suteu, Feb 20 2023
Name clarified and incorrect programs removed by Pontus von Brömssen, Sep 12 2023
a(18) from Max Alekseyev, Feb 24 2024
STATUS
approved
A060319 Smallest Fibonacci number with n distinct prime factors. +20
14
1, 2, 21, 610, 6765, 832040, 102334155, 190392490709135, 1548008755920, 23416728348467685, 2880067194370816120, 81055900096023504197206408605, 2706074082469569338358691163510069157, 5358359254990966640871840, 57602132235424755886206198685365216, 18547707689471986212190138521399707760 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..40
Ron Knott, Fibonacci numbers with tables of F(0)-F(500)
EXAMPLE
a(5) = F(30) = 832040 = 2^3 * 5 * 11 * 41 * 61.
MATHEMATICA
f[n_]:=Length@FactorInteger[Fibonacci[n]]; lst={}; Do[Do[If[f[n]==q, Print[Fibonacci[n]]; AppendTo[lst, Fibonacci[n]]; Break[]], {n, 280}], {q, 18}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
First /@ SortBy[#, Last] &@ Map[#[[1]] &, Values@ GroupBy[#, Last]] &@ Table[{#, PrimeNu@ #} &@ Fibonacci@ n, {n, 2, 200}] (* Michael De Vlieger, Feb 18 2017, Version 10 *)
CROSSREFS
Cf. A001605, A005478, A060320.
Row n=1 of A303218.
KEYWORD
nonn
AUTHOR
Labos Elemer, Mar 28 2001
STATUS
approved
A060320 Index of smallest Fibonacci number with exactly n distinct prime factors. +20
10
1, 3, 8, 15, 20, 30, 40, 70, 60, 80, 90, 140, 176, 120, 168, 180, 324, 252, 240, 378, 450, 432, 552, 360, 420, 690, 504, 880, 630, 600, 756, 720, 900, 792, 840, 1296, 1050, 1350, 1140, 1080, 1200 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
From Jon E. Schoenfield, Dec 28 2016: (Start)
a(42) = 1260, a(44) = 1320; for all other n > 40, a(n) > 1350.
These results were computed using data at the Blair Kelly site under Links at A022307. Note that the presence of incompletely factored Fibonacci numbers with indices as low as 1301 does not prevent the drawing of conclusions such as "a(44) = 1320" with certainly. Using F(1301) as an example, the compact table of Fibonacci results at the Kelly site indicates that F(1301) = p*q*r*c where p=6400921, q=14225131397, r=100794731109596201, and c is a 238-digit unfactored composite number. The complete factorization of every Fibonacci number up to F(1000) is explicitly given elsewhere on the site, and those results allow quick verification that a(n) <= 900 for all n in [0..34], so 1301 cannot be a term unless F(1301) has at least 35 distinct prime factors, which would require c to have at least 32 distinct prime factors, at least one of which would have to be less than ceiling(c^(1/32)) = 26570323, but trial division of c by every prime less than 26570323 shows that c has no prime factors that small. Thus, while A022307(1301) is unknown, it is certain that 1301 is not a term in this sequence. Similarly, making use of known factors, it can be proved that F(n) cannot have 44 or more distinct prime factors for any n < 1320, so since F(1320) has exactly 44 distinct prime factors, it is established that a(44) = 1320. (End)
LINKS
Table of n, a(n) for n=0..40.
Ron Knott, Fibonacci numbers with tables of F(0)-F(500)
Hisanori Mishima, Fibonacci numbers (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 480).
FORMULA
a(n) = min (k : A022307(k) = n).
EXAMPLE
n=9: F(80) = 23416728348467685 = 3 * 5 * 7 * 11 * 41 * 47 * 1601 * 2161 * 3041.
n=25: F(690) = 2^3 * 5 * 11 * 31 * 61 * 137 * 139 * 461 * 691 * 829 * 1151 * 1381 * 4831 * 5981 * 18077 * 28657 * 186301 * 324301 * 686551 * 1485571 * 4641631 * 117169733521 * 2441738887963981 * 3490125311294161 * 25013864044961447973152814604981 is the smallest Fibonacci number with exactly 25 distinct prime factors.
MATHEMATICA
First /@ SortBy[#, Last] &@ Map[First@ # &, Values@ GroupBy[#, Last]] &@ Table[{n - Boole[n == 2], #, PrimeNu@ #} &@ Fibonacci@ n, {n, 2, 300}] (* Michael De Vlieger, Feb 18 2017, Version 10 *)
Module[{ff=Table[{n, PrimeNu[Fibonacci[n]]}, {n, 1400}]}, Table[ SelectFirst[ ff, #[[2]]==k&], {k, 0, 40}]][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 28 2018 *)
PROG
(PARI) my(o=[], s); print1(1); for(n=1, 20, s=0; until( o[s]==n, #o<s++ && o=concat(o, omega(fibonacci(s))) ); print1(", "s))
CROSSREFS
Cf. A001605, A005478, A022307, A051694, A060319.
Row n=1 of A303217.
KEYWORD
nonn,more
AUTHOR
Labos Elemer, Mar 28 2001
EXTENSIONS
Corrected by Shyam Sunder Gupta, Jul 20 2002
Edited by M. F. Hasler, Nov 01 2012
a(35)-a(40) from Jon E. Schoenfield, Dec 28 2016
STATUS
approved
A048692 Triangle read by rows in which row n contains first n numbers with exactly n distinct prime factors. +20
9
2, 6, 10, 30, 42, 60, 210, 330, 390, 420, 2310, 2730, 3570, 3990, 4290, 30030, 39270, 43890, 46410, 51870, 53130, 510510, 570570, 690690, 746130, 870870, 881790, 903210, 9699690, 11741730, 13123110, 14804790, 15825810, 16546530, 17160990, 17687670, 223092870, 281291010, 300690390, 340510170, 358888530, 363993630, 380570190, 397687290, 406816410 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10011
Sean A. Irvine, Java program (github)
EXAMPLE
2;
6, 10;
30, 42, 60;
210, 330, 390, 420;
...
MATHEMATICA
f[n_] := Flatten[Table[ # [[1]]] & /@ FactorInteger[n]]; (* for n=7 *) Take[ Select[ Range[10^7], Length[f[ # ]] == 7 & ], 7]
Module[{nn=8, dpf=Table[{n, PrimeNu[n]}, {n, 2 10^7}]}, Flatten[Table[Select[dpf, #[[2]]==n&, n], {n, nn}], 1][[All, 1]]] (* The program generates the first 36 terms of the sequence. *) (* Harvey P. Dale, Sep 09 2022 *)
CROSSREFS
Cf. A002110 (first column).
Main diagonal gives A073329.
Extending the rows to give a square array, we get A125666.
KEYWORD
nonn,tabl
AUTHOR
Amarnath Murthy, Aug 20 2002
EXTENSIONS
Edited, corrected and extended by Robert G. Wilson v, Aug 22 2002
More terms from David A. Corneth, Jan 09 2021
STATUS
approved
A067024 Smallest prime p such that p+2 has exactly n distinct prime factors. +20
8
2, 13, 103, 1153, 15013, 255253, 4849843, 111546433, 4360010653, 100280245063, 5245694198743, 152125131763603, 7149881192889433, 421842990380476663, 16294579238595022363, 1106494163767990292293, 74135108972455349583763, 4632891063696575353839163, 278970415063349480483707693, 24012274383139350058948392193 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Sean A. Irvine, Table of n, a(n) for n = 1..58 (terms 1..38 from Michael S. Branicky)
Michael S. Branicky, Python program
Sean A. Irvine, Java program (github)
FORMULA
a(n) = Min_{p in A000040 ; A001221(p+2) = n}.
EXAMPLE
For n = 1,...,7 the factors of 2+a(n) are as follows: 2*2, 3*5, 3*5*7, 3*5*7*11, 3*5*7*11*13, 3*5*7*11*13*17, 3*5*7*11*13*17*19; i.e., a(n) = A002110(n+1)/2 which is prime for n = 2,...,7.
PROG
(Python) # see linked program
CROSSREFS
Cf. A000040, A001221, A002110, A053705, A067023.
KEYWORD
nonn
AUTHOR
Labos Elemer, Dec 29 2001
EXTENSIONS
a(8)-a(15) from Donovan Johnson, Jan 21 2009
a(16) and beyond from Michael S. Branicky, Feb 07 2023
STATUS
approved
A242621 Start of the least triple of consecutive squarefree numbers each of which has exactly n distinct prime factors. +20
8
2, 33, 1309, 27962, 3323705, 296602730, 41704979953 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
As the example of a(4)=27962 shows, "consecutive squarefree numbers" means consecutive elements of A005117, not necessarily consecutive integers that (additionally) are squarefree; this would be a more restrictive condition.
a(8) <= 102099792179229 because A093550 - 1 is an upper bound of the present sequence.
LINKS
Table of n, a(n) for n=1..7.
Daniel C. Mayer, Define an "m-triple" to consist of three consecutive squarefree positive integers, each with exactly m prime divisors, Number Theory group on LinkedIn.com
EXAMPLE
The two squarefree numbers following a(4)=27962, namely, 27965 and 27966, also have 4 prime divisors just as a(4).
CROSSREFS
See A242605-A242608 for triples of consecutive squarefree numbers with m=2,..,5 prime factors.
See A246470 for the quadruplet and A246548 for the 5-tuple versions of this sequence.
See A039833, A066509, A176167 and A192203 for triples of consecutive numbers which are squarefree and have m=2,..,5 prime factors.
KEYWORD
nonn
AUTHOR
M. F. Hasler, May 18 2014
EXTENSIONS
Edited and a(6)-a(7) added by Hans Havermann, Aug 27 2014
STATUS
approved
A156329 Smallest tetrahedral number with n distinct prime factors. +20
7
4, 10, 84, 1140, 7140, 125580, 2929290, 161226780, 2875585020, 32451432090, 9117204216120, 173092525291140, 12728365882372140, 6235727798083743960, 843456728066008506450, 68313970807402942762140, 7219596660397839675355860, 211677272048242059920400540 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..22
Eric Weisstein's World of Mathematics, Tetrahedral Numbers.
EXAMPLE
a(9) = 2875585020 = 2^2*3*5*7*11*17*19*41*47. 2875585020 is the smallest tetrahedral number with 9 distinct prime factors.
CROSSREFS
Cf. A000292.
KEYWORD
nonn
AUTHOR
Donovan Johnson, Feb 08 2009
EXTENSIONS
Edited by Donovan Johnson, Nov 24 2010
a(18) added (from b-file) by Jon E. Schoenfield, Dec 17 2022
STATUS
approved
A188342 Smallest odd primitive abundant number (A006038) having n distinct prime factors. +20
7
945, 3465, 15015, 692835, 22309287, 1542773001, 33426748355, 1635754104985, 114761064312895, 9316511857401385, 879315530560980695, 88452776289145528645, 2792580508557308832935, 428525983200229616718445, 42163230434005200984080045, 1357656019974967471687377449 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. For n=3, there are 8 such numbers: 945, 1575, 2205, 7425, 78975, 131625, 342225, 570375. See A188439.
a(14) <= 88452776289145528645. - Donovan Johnson, Mar 31 2011
a(15) <= 2792580508557308832935, a(16) <= 428525983200229616718445, a(17) <= 42163230434005200984080045. If these a(n) are squarefree and don't have a greatest prime factor more than 3 primes away from that of the preceding term, then these bounds are the actual values of a(n). The PARI code needs only fractions of a second to compute further bounds, which under the given hypotheses are the actual values of a(n). - M. F. Hasler, Jul 17 2016
It appears that the terms are squarefree for n >= 5, so they yield also the smallest term of A249263 with n factors; see A287581 for the largest such, and A287590 for the number of such terms with n factors. (For nonsquarefree odd abundant numbers, this seems to be known only for n = 3 and n = 4 prime factors (8 respectively 576 terms), cf. A188439.) - M. F. Hasler, May 29 2017
Comment from Don Reble, Jan 17 2023: (Start)
"If these a(n) are squarefree and don't have a greatest prime factor more than 3 primes away from that of the preceding term, then these bounds are the actual values of a(n)."
This conjecture is correct up to a(50). (End)
LINKS
Daniel Suteu, Table of n, a(n) for n = 3..27
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
H. N. Shapiro, Note on a theorem of Dickson, Bull Amer. Math. Soc. 55 (4) (1949), 450-452
EXAMPLE
From M. F. Hasler, Jul 17 2016: (Start)
945 = 3^3 * 5 * 7
3465 = 3^2 * 5 * 7 * 11
15015 = 3 * 5 * 7 * 11 * 13
692835 = 3 * 5 * 11 * 13 * 17 * 19 (n=6: gpf increases by 2 primes)
22309287 = 3 * 7 * 11 * 13 * 17 * 19 * 23
1542773001 = 3 * 7 * 11 * 17 * 19 * 23 * 29 * 31
33426748355 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31
1635754104985 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 37 * 41 (here too)
114761064312895 = 5 * 7 * 11 * 13 * 17 * 23 * 29 * 31 * 37 * 41 * 43
9316511857401385 = 5 * 7 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47
879315530560980695 = 5 * 7 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 53 * 59 * 61 (n=13: gpf increases for the first time by 3 primes) (End)
MATHEMATICA
PrimAbunQ[n_] := Module[{x, y},
y = Most[Divisors[n]]; x = DivisorSigma[1, y];
DivisorSigma[1, n] > 2 n && AllTrue[x/y, # <= 2 &]];
Table[k = 1;
While[! PrimAbunQ[k] || Length[FactorInteger[k][[All, 1]]] != n,
k += 2]; k, {n, 3, 6}] (* Robert Price, Sep 26 2019 *)
PROG
(PARI) A188342=[0, 0, 945, 3465]; a(n, D(n)=n\6+1)={while(n>#A188342, my(S=#A188342, T=factor(A188342[S])[, 1], M=[primepi(T[1]), primepi(T[#T])+D(S++)], best=prime(M[2])^S); forvec(v=vector(S, i, M), best>(T=prod(i=1, #v, prime(v[i]))) && (S=prod(i=1, #v, prime(v[i])+1)-T*2)>0 && S*prime(v[#v])<T*2 && best=T, 2); A188342=concat(A188342, best)); A188342[n]} \\ Assuming a(n) squarefree for n>4, search is exhaustive within the limit primepi(gpf(a(n))) <= primepi(gpf(a(n-1)))+D(n), with D(n) given as optional 2nd arg. - M. F. Hasler, Jul 17 2016
(PARI)
generate(A, B, n) = A=max(A, vecprod(primes(n+1))\2); (f(m, p, j) = my(list=List()); if(sigma(m) > 2*m, return(list)); forprime(q=p, sqrtnint(B\m, j), my(v=m*q); while(v <= B, if(j==1, if(v>=A && sigma(v) > 2*v, my(F=factor(v)[, 1], ok=1); for(i=1, #F, if(sigma(v\F[i], -1) > 2, ok=0; break)); if(ok, listput(list, v))), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=vecprod(primes(n+1))\2, y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 10 2024
CROSSREFS
Cf. A006038, A275449.
KEYWORD
nonn
AUTHOR
T. D. Noe, Mar 28 2011
EXTENSIONS
a(8)-a(12) from Donovan Johnson, Mar 29 2011
a(13) from Donovan Johnson, Mar 31 2011
a(14)-a(17) confirmed and a(18) from Daniel Suteu, Feb 10 2024
STATUS
approved
A188439 Irregular triangle of odd primitive abundant numbers (A006038) in which row n has numbers with n distinct prime factors. +20
7
945, 1575, 2205, 7425, 78975, 131625, 342225, 570375, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 16695, 18585, 19215, 21105, 22365, 22995, 24885, 26145, 28035, 28215, 29835 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
The initial row has 8 terms. Row n begins with A188342(n). Dickson proves that each row has a finite number of terms. He lists the first two rows in factored form in his paper. However, as Ferrier and Herzog report, Dickson's tables have many errors. There are 576 odd primitive abundant numbers (OPAN) having 4 distinct prime factors, the last of which is 3^10 5^5 17^4 251^2 = 970969744245403125. The next row, for 5 distinct prime factors, has over 100000 terms.
If the prime factors were counted with multiplicity, then the table would start with row 5, having 121 terms: (945, 1575, 2205, 3465, 4095, ..., 430815, 437745, 442365). Row 6 would start (7425, 28215, 29835, 33345, 34155, ...), and row 7, (81081, 121095, 164835, 182655, 189189, ...). - M. F. Hasler, Jul 27 2016 [See A287646.]
LINKS
T. D. Noe, Rows n = 3..4, flattened
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
A. Ferrier, Table errata 176, MTAC 4 (1950), 222.
Fritz Herzog, Table Errata 571, Math. Comp. 34 (1980), 652.
T. D. Noe, 576 odd primitive abundant numbers, factored
EXAMPLE
From M. F. Hasler, Jul 27 2016: (Start)
Row 3: 945, 1575, 2205, 7425, 78975, 131625, 342225, 570375;
Row 4: 3465, 4095, 5355, ...(571 more)..., 249450402403828125, 970969744245403125;
Row 5: 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, ...
Row 6: 692835, 838695, 937365, 1057485, 1130415, 1181895, 1225785, 1263405, ...
Row 7: 22309287, 28129101, 30069039, 34051017, 35888853, 36399363, ...
The first column is A188342 = (945, 3465, 15015, 692835, 22309287, ...) (End)
CROSSREFS
Row lengths are A303933.
Cf. A006038 (all OPAN), A188342 (first column of this table), A287646 (variant where row n contains all OPAN with n prime factors counted with multiplicity).
KEYWORD
nonn,tabf
AUTHOR
T. D. Noe, Mar 31 2011
STATUS
approved
A358862 a(n) is the smallest n-gonal number with exactly n distinct prime factors. +20
7
66, 44100, 11310, 103740, 3333330, 185040240, 15529888374, 626141842326, 21647593547580, 351877410344460, 82634328555218440, 2383985537862979050, 239213805711830629680 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
The corresponding indices of n-gonal numbers are 11, 210, 87, 228, 1155, 7854, 66612, 395646, 2193303, ...
LINKS
Table of n, a(n) for n=3..15.
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Polygonal Number
EXAMPLE
a(3) = 66, because 66 is a triangular number with 3 distinct prime factors {2, 3, 11} and this is the smallest such number.
MATHEMATICA
Table[SelectFirst[PolygonalNumber[n, Range[400000]], PrimeNu[#]==n&], {n, 3, 10}] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Sep 09 2023 *)
PROG
(PARI) a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(n*k - n - 2*k + 4))\2); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 04 2022
(PARI)
omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A && ispolygonal(v, k), listput(list, v)), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(n < 3, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 04 2022
CROSSREFS
Cf. A001221, A076551, A156236, A156237, A156238, A156239, A358863, A358864, A358865.
KEYWORD
nonn,more
AUTHOR
Ilya Gutkovskiy, Dec 03 2022
EXTENSIONS
a(12)-a(15) from Daniel Suteu, Dec 04 2022
STATUS
approved
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Last modified April 19 02:28 EDT 2024. Contains 371782 sequences. (Running on oeis4.)