A054992 -id:A054992 - cp's OEIS frontend

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

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

A046798 Number of divisors of 2^n + 1.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 4, 4, 2, 8, 6, 4, 4, 4, 8, 12, 2, 4, 16, 4, 4, 12, 8, 4, 8, 16, 16, 20, 4, 8, 48, 4, 4, 24, 16, 32, 16, 8, 16, 12, 4, 8, 64, 4, 8, 64, 32, 8, 8, 8, 64, 48, 8, 8, 64, 48, 8, 24, 8, 16, 16, 4, 32, 64, 4, 64, 64, 8, 12, 24, 96, 8, 32, 8, 32, 96, 16, 64, 768, 4, 8, 192, 32, 64

Offset: 0

Labos Elemer

nonn

a(n) is odd iff n = 3, as a consequence of the Catalan-Mihăilescu theorem. - Bernard Schott, Oct 05 2021

			a(7)=4, because 2^7 + 1 = 129 has 4 divisors.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..500 from T. D. Noe, terms 501..1062 from Amiram Eldar, term 1108 from Tyler Busby)
Wikipedia, Catalan's conjecture.

Cf. A000005, A000051, A002587, A002589, A046801, A054992, A057957, A059886, A274906, A344897.

[NumberOfDivisors(2^n+1): n in [0..100]]; // Vincenzo Librandi, Feb 05 2018

a:= n-> numtheory[tau](2^n+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 23 2021

A046798[n_IntegerQ]:=DivisorSigma[0,1+2^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
DivisorSigma[0, 1 + 2^#] & /@ Range[0, 83] (* Jayanta Basu, Jun 29 2013 *)
Table[DivisorSigma[0, 2^n + 1], {n, 0, 100}] (* Vincenzo Librandi, Feb 05 2018 *)

a(n) = numdiv(2^n+1); \\ Michel Marcus, Mar 18 2017

from sympy.ntheory import divisor_count
def A046798(n): return divisor_count(2**n + 1) # Indranil Ghosh, Mar 18 2017

a(n) = A000005(A000051(n)). - Michel Marcus, Mar 18 2017

A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 3, 2, 2, 2, 2, 2, 5, 3, 5, 3, 3, 4, 7, 5, 4, 3, 2, 4, 8, 4, 5, 3, 5, 3, 7, 4, 3, 7, 2, 4, 9, 4, 5, 6, 4, 3, 10, 4, 3, 7, 4, 4, 12, 4, 4, 9, 4, 7, 8, 4, 2, 6, 10, 5, 6, 5, 4, 6, 3, 3, 12, 3, 6, 8, 2, 4, 10, 11, 3, 5, 4, 7, 11, 6, 12, 7, 4, 9, 11, 3, 7, 8, 8, 3, 8, 4, 4, 11, 6, 4, 8, 4, 6, 8, 4, 5, 13

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

2^(a(2n)-1)-1 predicts the number of pair-solutions of even length L for AB = A^2 + B^2. For instance, with length 18 we have 10^18 + 1 = 101*9901*999999000001 or 3 divisors F which when put into the Mersenne formula 2^(F-1)-1 yields 3 pairs (see reference 'Puzzle 104' for details).

Max Alekseyev, Table of n, a(n) for n = 1..331 (first 310 terms from Ray Chandler)
Makoto Kamada, Factorizations of 100...001.
Carlos Rivera, Puzzle 104, The Prime Puzzles & Problems Connection.
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): this sequence (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A003021, A001271, A046053, A001562, A057951, A119704, A344897.

PrimeOmega[10^Range[100]+1] (* Harvey P. Dale, May 02 2021 *)

a(n)=bigomega(10^n+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A057951(2n) - A057951(n). - T. D. Noe, Jun 19 2003

A057957 Number of prime factors of 4^n - 1 (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 3, 4, 6, 6, 4, 7, 3, 6, 7, 5, 3, 10, 3, 8, 8, 7, 4, 10, 7, 7, 9, 8, 6, 13, 3, 7, 9, 7, 9, 14, 5, 7, 8, 10, 5, 14, 5, 10, 13, 9, 6, 13, 5, 14, 11, 10, 6, 15, 12, 11, 9, 9, 6, 17, 3, 8, 14, 9, 9, 15, 5, 11, 9, 16, 6, 19, 6, 10, 14, 11, 10, 18, 5, 13, 16, 10, 8, 19, 7, 10, 11

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1128 (first 603 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), this sequence (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A024036, A054992, A085029, A274906.

PrimeOmega/@(4^Range[90]-1) (* Harvey P. Dale, Dec 31 2018 *)

Mobius transform of A085029. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024036(n)) = A046051(2*n). - Amiram Eldar, Feb 01 2020

A057935 Number of prime factors of 9^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 3, 3, 4, 3, 4, 2, 4, 3, 4, 6, 4, 4, 6, 2, 4, 4, 4, 5, 7, 5, 4, 4, 8, 4, 5, 6, 4, 7, 5, 2, 6, 5, 9, 8, 5, 6, 7, 5, 5, 10, 7, 6, 9, 4, 4, 6, 9, 6, 8, 7, 6, 9, 8, 9, 9, 5, 3, 11, 6, 4, 11, 6, 8, 9, 9, 8, 6, 9, 5, 6, 6, 6, 13, 4, 8, 7, 5, 4, 7, 6, 5, 11, 8, 5, 8, 7, 4, 11, 7, 9, 9, 5, 9, 7, 5, 6, 10, 7, 6

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..345 (first 329 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), this sequence (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A002592, A046053, A001222, A057941, A057952, A062396.

f:=func; [f(9^n + 1):n in [1..100]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[Table[9^n + 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)

a(n) = A057952(2n) - A057952(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A062396(n)) = A057941(2*n). - Amiram Eldar, Feb 02 2020

A057941 Number of prime factors of 3^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 3, 3, 5, 4, 4, 3, 3, 4, 6, 2, 5, 4, 4, 3, 7, 4, 3, 6, 5, 4, 7, 4, 5, 6, 4, 2, 7, 4, 5, 4, 5, 4, 8, 5, 4, 7, 3, 5, 10, 4, 5, 4, 5, 8, 9, 4, 4, 5, 7, 6, 8, 4, 4, 7, 4, 5, 13, 2, 5, 6, 4, 5, 9, 9, 7, 8, 4, 5, 12, 6, 6, 7, 5, 5, 12, 5, 6, 10, 9, 7, 11, 6, 5, 9, 8, 4, 9, 4, 8, 6, 5, 9, 14, 6, 4

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..691 (first 658 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), this sequence (b=3), A054992 (b=2).

Cf. A001222, A007658, A034472, A057958, A074476.

PrimeOmega[3^Range[110]+1] (* Harvey P. Dale, Jun 20 2015 *)

a(n)=bigomega(n^3+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A057958(2n) - A057958(n) - T. D. Noe, Jun 19 2003

a(n) = A001222(A034472(n)). - Amiram Eldar, Feb 01 2020

A057936 Number of prime factors of 8^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 4, 2, 4, 4, 4, 3, 6, 6, 5, 4, 4, 6, 7, 3, 6, 6, 5, 4, 7, 6, 5, 5, 7, 10, 10, 5, 5, 11, 5, 3, 9, 9, 11, 6, 7, 8, 7, 6, 7, 10, 6, 7, 12, 8, 7, 7, 7, 14, 11, 5, 6, 10, 12, 8, 9, 8, 8, 8, 4, 9, 13, 4, 11, 12, 8, 9, 8, 15, 8, 8, 6, 10, 12, 8, 12, 17, 6, 7, 15, 10, 9, 12, 12, 10, 11, 8, 8, 18, 12

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502 (first 354 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), this sequence (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A057953, A062395, A274905.

f:=func; [f(8^n + 1):n in [1..110]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[8^Range[100]+1] (* Harvey P. Dale, Dec 16 2014 *)

a(n) = A057953(2n) - A057953(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A062395(n)) = A054992(3*n). - Amiram Eldar, Feb 02 2020

A057937 Number of prime factors of 7^n + 1 (counted with multiplicity).

Original entry on oeis.org

3, 3, 4, 2, 5, 5, 5, 3, 5, 6, 5, 5, 5, 4, 7, 3, 4, 6, 5, 4, 8, 6, 4, 4, 6, 5, 6, 6, 4, 10, 6, 4, 8, 6, 11, 8, 5, 4, 9, 6, 7, 7, 10, 5, 12, 7, 4, 6, 12, 11, 8, 5, 7, 9, 13, 8, 9, 7, 7, 9, 4, 6, 14, 4, 9, 12, 6, 5, 8, 10, 5, 9, 6, 4, 11, 7, 12, 10, 7, 6, 10, 6, 5, 12, 8, 6, 8, 5, 8, 14, 11, 6, 9, 6, 11

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..387 (first 372 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), this sequence (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A034491, A057954, A227575.

f:=func; [f(7^n + 1):n in [1..110]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[Table[7^n + 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)

a(n) = A057954(2n) - A057954(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A034491(n)). - Amiram Eldar, Feb 02 2020

A057939 Number of prime factors of 5^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 4, 2, 3, 3, 4, 3, 6, 4, 5, 3, 4, 3, 7, 3, 4, 5, 5, 4, 10, 4, 4, 4, 5, 5, 10, 3, 4, 7, 5, 4, 9, 6, 7, 6, 5, 4, 8, 5, 6, 6, 6, 3, 10, 3, 5, 5, 7, 7, 10, 5, 5, 6, 7, 7, 9, 3, 6, 6, 6, 4, 16, 4, 8, 7, 3, 7, 9, 7, 5, 6, 5, 5, 13, 5, 9, 10, 6, 6, 14, 6, 5, 7, 9, 5, 9, 7, 5, 12, 8, 4, 10, 6, 9, 7, 7, 7, 12

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..471 (first 451 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), this sequence (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A034474, A057956, A074478.

PrimeOmega[5^Range[100]+1] (* Harvey P. Dale, Nov 27 2013 *)

a(n) = A057956(2n) - A057956(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A034474(n)). - Amiram Eldar, Feb 01 2020

A057940 Number of prime factors of 4^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 1, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 3, 5, 3, 7, 3, 6, 3, 3, 4, 5, 2, 6, 4, 7, 5, 5, 4, 10, 3, 5, 5, 5, 4, 11, 2, 4, 3, 6, 6, 9, 2, 4, 6, 7, 5, 8, 3, 7, 6, 6, 4, 10, 2, 10, 7, 6, 4, 8, 4, 6, 7, 5, 2, 14, 4, 9, 5, 4, 4, 10, 4, 6, 8, 11, 4, 8, 3, 4, 8, 11, 4, 9, 5, 10, 4, 9, 8, 12, 6

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..583 (first 531 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), this sequence (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A052539, A057957, A274903.

with(numtheory); A057940:=n->bigomega(4^n + 1); seq(A057940(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

Table[PrimeOmega[4^n + 1], {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n) = A057957(2n) - A057957(n). - T. D. Noe, Jun 19 2003
a(n) = Omega(4^n + 1) = A001222(A052539(n)). - Wesley Ivan Hurt, Jan 28 2014
a(n) = A054992(2*n). - Amiram Eldar, Feb 01 2020

cp's OEIS Frontend

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

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A046798 Number of divisors of 2^n + 1.

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A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

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A057957 Number of prime factors of 4^n - 1 (counted with multiplicity).

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A057935 Number of prime factors of 9^n + 1 (counted with multiplicity).

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A057941 Number of prime factors of 3^n + 1 (counted with multiplicity).

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A057936 Number of prime factors of 8^n + 1 (counted with multiplicity).

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A057937 Number of prime factors of 7^n + 1 (counted with multiplicity).

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