A046800 -id:A046800 - cp's OEIS frontend

A344932 Values of n where A046800(n) produces a record.

Original entry on oeis.org

0, 2, 4, 8, 12, 20, 24, 36, 48, 60, 72, 120, 144, 180, 288, 300, 360, 420, 540, 660, 720, 840, 1080

Offset: 1

Chai Wah Wu, Jun 02 2021

Cf. A046800, A177710.

A344933 Record values of A046800.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 15, 17, 21, 24, 25, 29, 34, 36, 37, 39, 46, 51

Offset: 1

Chai Wah Wu, Jun 02 2021

Cf. A046800, A344932.

DeleteDuplicates[Join[{0},PrimeNu/@(2^Range[200]-1)],GreaterEqual] (* The program generates the first 14 terms of the sequence. *) (* Harvey P. Dale, Dec 24 2022 *)

A046051 Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, 1, 6, 4, 4, 2, 7, 3, 3, 3, 6, 3, 7, 1, 5, 4, 3, 4, 10, 2, 3, 4, 8, 2, 8, 3, 7, 6, 4, 3, 10, 2, 7, 5, 7, 3, 9, 6, 8, 4, 6, 2, 13, 1, 3, 7, 7, 3, 9, 2, 7, 4, 9, 3, 14, 3, 5, 7, 7, 4, 8, 3, 10, 6, 5, 2, 14, 3, 5, 6, 10, 1, 13, 5, 9, 3, 6, 5, 13, 2, 5, 8

Offset: 1

Eric W. Weisstein

nonn

Length of row n of A001265.

			a(4) = 2 because 2^4 - 1 = 15 = 3*5.
From _Gus Wiseman_, Jul 04 2019: (Start)
The sequence of Mersenne numbers together with their prime indices begins:
        1: {}
        3: {2}
        7: {4}
       15: {2,3}
       31: {11}
       63: {2,2,4}
      127: {31}
      255: {2,3,7}
      511: {4,21}
     1023: {2,5,11}
     2047: {9,24}
     4095: {2,2,3,4,6}
     8191: {1028}
    16383: {2,14,31}
    32767: {4,11,36}
    65535: {2,3,7,55}
   131071: {12251}
   262143: {2,2,2,4,8,21}
   524287: {43390}
  1048575: {2,3,3,5,11,13}
(End)

Sean A. Irvine, Table of n, a(n) for n = 1..1206 (terms 1..500 from T. D. Noe)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Alex Kontorovich, Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Number

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), this sequence (b=2).
Cf. A000043, A000668, A001348, A054988, A054989, A054990, A054991, A054992, A085021.
Cf. A000225, A001221, A001222, A046800, A049093, A059305, A325610, A325611, A325612, A325625.

a[q_] := Module[{x, n}, x=FactorInteger[2^n-1]; n=Length[x]; Sum[Table[x[i][2], {i, n}][j], {j, n}]]
a[n_Integer] := PrimeOmega[2^n - 1]; Table[a[n], {n,200}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(2^n-1) \\ Charles R Greathouse IV, Apr 01 2013

Mobius transform of A085021. - T. D. Noe, Jun 19 2003

a(n) = A001222(A000225(n)). - Michel Marcus, Jun 06 2019

A016047 Smallest prime factor of Mersenne numbers 2^p-1, where p is prime.

Original entry on oeis.org

3, 7, 31, 127, 23, 8191, 131071, 524287, 47, 233, 2147483647, 223, 13367, 431, 2351, 6361, 179951, 2305843009213693951, 193707721, 228479, 439, 2687, 167, 618970019642690137449562111, 11447, 7432339208719, 2550183799, 162259276829213363391578010288127

Offset: 1

Robert G. Wilson v

"Mersenne numbers", here, means A001348. Compare to sequence A049479, where "Mersenne numbers" is used in the sense of A000225. - Lekraj Beedassy, Jun 11 2009
Submitted new b-file withdrawing last three terms previously submitted.  I had, before submitting that b-file, checked that the smallest known factors of incompletely factored Mersenne numbers was less than the known trial factoring limits recorded by Will Edgington in his LowM.txt file which can be found on his Mersenne page, (see link above.)  I have now discovered that I inadvertently omitted the purported a(203) from that check. - Daran Gill, Apr 05 2013
The would-be a(203) corresponds to 2^1237-1 which is currently P70*C303. Trial factoring has only been done to 60 bits, and since its difficulty doubles whenever the bit length is incremented by one, it cannot exhaustively search the space smaller than the sole known 70-digit (231-bit) factor. Probabilistic ECM testing indicates only that it is extremely unlikely that there is any undiscovered factor with digit-size smaller than the high fifties. See GIMPS links. - Gord Palameta, Aug 16 2018

Daran Gill, Table of n, a(n) for n = 1..202 (first 95 terms from T. D. Noe)
C. K. Caldwell, Mersenne Primes
Will Edgington, Mersenne Page
GIMPS, PrimeNet Known Factors of Mersenne Numbers (discovered recently for p < 2000), Exponent status of 2^1237 - 1, ECM testing status of 2^1237 - 1
Brady Haran and Matt Parker, How they found the World's Biggest Prime Number - Numberphile, Numberphile video (2016).

Cf. A000668 (a subsequence), A003260, A001348, A020639, A046800.

a:= n-> min(numtheory[factorset](2^ithprime(n)-1)):
seq(a(n), n=1..28);  # Alois P. Heinz, Oct 01 2024

a = {}; Do[If[PrimeQ[n], w = 2^n - 1; c = FactorInteger[w]; b = c[[1]][[1]]; AppendTo[a, b]], {n, 2, 100}]; a (* Artur Jasinski, Dec 11 2007 *)

forprime(p=2,150,print1(factor(2^p-1)[1,1],", "))

a(n) = A020639(A001348(n)). - Alois P. Heinz, Oct 01 2024

A366707 Number of distinct prime divisors of 12^n - 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 3, 8, 3, 6, 6, 9, 3, 9, 2, 7, 5, 5, 4, 12, 4, 7, 6, 10, 5, 13, 5, 11, 7, 6, 9, 14, 3, 6, 7, 13, 4, 13, 5, 11, 12, 8, 3, 18, 5, 10, 6, 12, 7, 16, 7, 13, 7, 8, 4, 18, 4, 8, 8, 13, 8, 16, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 10, 17, 4

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A024140, A001221, A046800, A133801, A102347, A250288, A252170, A366708, A366709, A366681.

for(n = 1, 100, print1(omega(12^n - 1), ", "))

a(n) = omega(12^n-1) = A001221(A024140(n)).

A182590 Number of distinct prime factors of 2^n - 1 of the form k*n + 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 3, 2, 1, 2, 2, 1, 2, 3, 3, 1, 2, 1, 2, 2, 3, 2, 2, 3, 2, 2, 4, 3, 3, 2, 3, 3, 3, 1, 4, 4, 3, 3, 2, 3, 2, 3, 5, 2, 2, 1, 2, 3, 4, 2, 3, 2, 3, 1, 4, 3, 3, 3, 4, 5, 3, 1, 5, 3, 2, 3, 4, 2, 3, 2, 4, 3

Offset: 2

Seppo Mustonen, Nov 22 2010

nonn

From Thomas Ordowski, Sep 08 2017: (Start)
By Bang's theorem, a(n) > 0 for all n > 1, see A186522.
Primes p such that a(p) = 1 are the Mersenne exponents A000043.
Composite numbers m for which a(m) = 1 are A292079.
a(n) >= A086251(n), where equality is for all prime numbers and for some composite numbers (among others for all odd prime powers p^k with k > 1).
Theorem: if n is prime, then a(n) = A046800(n).
Conjecture: if a(n) = A046800(n), then n is prime.
Problem: is a(n) < A046800(n) for every composite n? (End)

			For n=10 the prime factors of 2^n - 1 = 1023 are 3, 11 and 31, and 11 = n+1, 31 = 3n+1. Thus a(10)=2.

Charles R Greathouse IV, Table of n, a(n) for n = 2..1200 (terms 2..200 from Seppo Mustonen, terms 201..786 from Michel Marcus)
S. Mustonen, On prime factors of numbers m^n+-1, 2010.
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

Cf. A046800, A086251, A186522.

m = 2; n = 2; nmax = 200;
While[n <= nmax, {l = FactorInteger[m^n - 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]

a(n) = my(f = factor(2^n-1)); sum(k=1, #f~, ((f[k,1]-1) % n)==0); \\ Michel Marcus, Sep 10 2017

Name edited by Thomas Ordowski, Sep 19 2017

A366681 Number of distinct prime divisors of 11^n - 1.

Original entry on oeis.org

2, 3, 4, 4, 3, 6, 4, 5, 5, 5, 4, 9, 4, 6, 6, 7, 3, 8, 3, 7, 9, 9, 5, 12, 6, 8, 6, 10, 4, 11, 5, 9, 9, 7, 7, 12, 6, 8, 12, 10, 9, 13, 4, 12, 8, 10, 5, 18, 7, 10, 9, 10, 6, 11, 9, 15, 7, 8, 5, 16, 5, 10, 15, 12, 7, 19, 6, 12, 10, 15, 7, 18, 3, 9, 13, 11, 8, 20

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..316

Cf. A024127, A001221, A274910, A366682, A366683, A366684, A366685.

Cf. A046800, A133801, A102347, A366707.

for(n = 1, 100, print1(omega(11^n - 1), ", "))

a(n) = omega(11^n-1) = A001221(A024127(n)).

A003260 Largest prime factor of n-th Mersenne number (A001348(n)).

Original entry on oeis.org

3, 7, 31, 127, 89, 8191, 131071, 524287, 178481, 2089, 2147483647, 616318177, 164511353, 2099863, 13264529, 20394401, 3203431780337, 2305843009213693951

Offset: 1

N. J. A. Sloane

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michel Marcus and Gord Palameta, Table of n, a(n) for n = 1..197 (first 95 terms from T. D. Noe, derived from Brillhart et al.)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
C. K. Caldwell, Mersenne primes
P. Erdős and T. N. Shorey, On the greatest prime factor of 2^p-1 for a prime p and other expressions, Acta Arith. 30:3 (1976), pp. 257-265.
C. L. Stewart, The greatest prime factor of a^n - b^n, Acta Arith. 26 (1975), pp. 427-433.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000668 (a subsequence), A001348, A016047, A046800.

a[n_] := FactorInteger[ 2^Prime[n] - 1 ][[-1, 1]]; Table[ a[n], {n, 1, 18}] (* Jean-François Alcover, Dec 20 2011 *)

a(n)=my(f=factor(2^prime(n)-1)[,1]);f[#f] \\ Charles R Greathouse IV, Dec 05 2012

Let p = prime(n). Erdős & Shorey show that a(n) >= kp log p for some effectively computable k >= 1. (Presumably k can be chosen as 7/log 27.) - Charles R Greathouse IV, Dec 05 2012

A086251 Number of primitive prime factors of 2^n - 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3

Offset: 1

T. D. Noe, Jul 14 2003

A prime factor of 2^n - 1 is called primitive if it does not divide 2^r - 1 for any r < n. Equivalently, p is a primitive prime factor of 2^n - 1 if ord(2,p) = n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors.
Number of odd primes p such that A002326((p-1)/2) = n. Number of occurrences of number n in A014664. - Thomas Ordowski, Sep 12 2017
The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020

			a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.

Table of n, a(n) for n = 1..1206
Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.

Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]

a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017

a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ Jeppe Stig Nielsen, Sep 01 2020

a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
a(n) <= A182590(n). - Thomas Ordowski, Sep 14 2017
a(n) = A001221(A064078(n)). - Thomas Ordowski, Oct 26 2017

Terms to a(500) in b-file from T. D. Noe, Nov 11 2010
Terms a(501)-a(1200) in b-file from Charles R Greathouse IV, Sep 14 2017
Terms a(1201)-a(1206) in b-file from Max Alekseyev, Sep 11 2022

A366620 Number of distinct prime divisors of 6^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 4, 5, 3, 7, 3, 5, 5, 6, 5, 7, 3, 8, 4, 5, 5, 9, 4, 7, 6, 8, 2, 10, 3, 9, 6, 8, 6, 13, 6, 6, 6, 11, 3, 9, 5, 9, 10, 8, 4, 13, 5, 8, 9, 11, 4, 11, 6, 13, 7, 6, 4, 19, 4, 5, 10, 12, 8, 12, 3, 11, 8, 16, 2, 18, 5, 10, 10, 9, 6, 15, 4, 16, 8

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..430

Cf. A024062, A001221, A057955, A059888, A274907, A366621, A366622, A366623.

Cf. A046800, A133801, A366604, A366611, A366632, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(6^n - 1), ", "))

a(n) = omega(6^n-1) = A001221(A024062(n)).

cp's OEIS Frontend

A344932 Values of n where A046800(n) produces a record.

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A344933 Record values of A046800.

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A046051 Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity).

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A016047 Smallest prime factor of Mersenne numbers 2^p-1, where p is prime.

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A366707 Number of distinct prime divisors of 12^n - 1.

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A182590 Number of distinct prime factors of 2^n - 1 of the form k*n + 1.

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A366681 Number of distinct prime divisors of 11^n - 1.

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A003260 Largest prime factor of n-th Mersenne number (A001348(n)).

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A086251 Number of primitive prime factors of 2^n - 1.

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