A054988 -id:A054988 - cp's OEIS frontend

A056160 Sum of A054988 and A054989.

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

Offset: 1

Arne Ring (arne.ring(AT)epost.de), Aug 01 2000

Prime divisors are counted with multiplicity.

If "2" were a cluster point of this sequence it would follow that there are infinitely many twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..98

Cf. A054988, A054989.

a(n) = my(P=prod(k=1, n, prime(k))); bigomega(P+1) + bigomega(P-1); \\ Michel Marcus, Mar 07 2022

a(36)-a(81) from Charles R Greathouse IV, May 07 2011

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

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 4, 3, 2, 2, 2, 3, 4, 1, 2, 4, 2, 2, 4, 3, 2, 3, 4, 4, 6, 2, 3, 6, 2, 2, 5, 4, 5, 4, 3, 4, 4, 2, 3, 6, 2, 3, 7, 5, 3, 3, 3, 7, 6, 3, 3, 6, 6, 3, 5, 3, 4, 4, 2, 5, 7, 2, 6, 6, 3, 4, 5, 7, 3, 5, 3, 5, 7, 4, 6, 10, 2, 3, 10, 5, 6, 5, 4, 5, 5, 4, 4, 11, 6, 2, 5, 4, 5, 3, 5, 6, 9, 6, 2, 9, 3

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

nonn

The length of row n in A001269.

			a(3) = 2 because 2^3 + 1 = 9 = 3*3.

Max Alekseyev, Table of n, a(n) for n = 1..1128
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), A057941 (b=3), this sequence (b=2).
Cf. A000051, A002586, A002587, A003260, A001222, A001269, A001348, A054988, A054989, A054990, A054991, A000978.
Cf. A046051 (number of prime factors of 2^n-1).
Cf. A086257 (number of primitive prime factors).

a[n_] := Module[{x=FactorInteger[2^n+1]}, Sum[x[[i]][[2]], {i, Length[x]}]]
A054992[n_Integer] := PrimeOmega[2^n + 1]; Table[A054992[n], {n,103}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(2^n+1) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = A046051(2n) - A046051(n). - T. D. Noe, Jun 18 2003

a(n) = A001222(A000051(n)). - Amiram Eldar, Oct 04 2019

Extended by Patrick De Geest, Oct 01 2000
Terms to a(500) in b-file from T. D. Noe, Nov 10 2007
Deleted duplicate (and broken) Wagstaff link. - N. J. A. Sloane, Jan 18 2019
a(500)-a(1062) in b-file from Amiram Eldar, Oct 04 2019
a(1063)-a(1128) in b-file from Max Alekseyev, Jul 15 2023, Mar 15 2025

A054991 Number of prime divisors of n! - 1 (counted with multiplicity).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 1, 2, 3, 2, 4, 1, 2, 1, 5, 2, 3, 3, 3, 2, 4, 3, 2, 2, 3, 2, 2, 4, 5, 1, 3, 1, 1, 2, 3, 2, 5, 1, 4, 2, 4, 4, 7, 4, 5, 5, 2, 4, 3, 2, 5, 5, 4, 6, 6, 5, 6, 5, 2, 3, 4, 4, 5, 4, 6, 4, 7, 2, 6, 5, 5, 3, 4, 5, 7, 3, 5, 4, 2, 4, 4, 4, 4, 6, 2, 3, 4

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

nonn

The series is related to the product of primes and the "proof" of the existence of infinite many prime twins.

			a(2)=0 because 2! - 1 = 1 (and this is not a prime number) a(5)=2 because 5! -1 = 119 = 7 * 17

Amiram Eldar, Table of n, a(n) for n = 1..135
R. G. Wilson v, Explicit factorizations
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences

Cf. A033312, A054988, A054989, A054990, A054992.

a[q_] := Module[{x, n}, x=FactorInteger[q!-1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
A054991[n_Integer] := PrimeOmega[n! - 1]; A054991[1] = 0; Table[A054991[n], {n,2,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

More terms from Robert G. Wilson v, Mar 24 2001

More terms from Amiram Eldar, Oct 03 2019

A054989 Number of prime divisors of -1 + (product of first n primes).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 3, 2, 2, 4, 1, 2, 3, 3, 2, 3, 3, 2, 2, 4, 3, 1, 2, 2, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 4, 4, 2, 3, 3, 2, 4, 3, 4, 2, 4, 4, 7, 4, 3, 3, 4, 4, 3, 3, 1, 3, 1, 4, 3, 5, 5, 4, 4, 6, 5, 5, 3, 4, 3, 4, 4, 3, 4, 2, 3, 4

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

			a(4)=2 because 2*3*5*7 - 1 = 209 = 11*19

Amiram Eldar, Table of n, a(n) for n = 1..99
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A054988, A054990, A054991, A054992, A057588.

a[q_] := Module[{x, n}, x=FactorInteger[Product[Table[Prime[i], {i, q}][[j]], {j, q}]-1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
PrimeOmega[#] & /@ (FoldList[Times, Prime[Range[81]]] - 1) (* Harvey P. Dale, Mar 11 2017 *)

More terms from Robert G. Wilson v, Mar 24 2001
a(42)-a(81) from Charles R Greathouse IV, May 07 2011
a(82)-a(87) from Amiram Eldar, Oct 03 2019

A054990 Number of prime divisors of n! + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 3, 5, 3, 6, 2, 2, 3, 3, 4, 2, 2, 2, 1, 2, 3, 5, 4, 4, 5, 2, 5, 6, 1, 2, 4, 7, 1, 3, 4, 3, 3, 3, 4, 2, 5, 5, 6, 4, 4, 2, 2, 4, 3, 4, 2, 4, 4, 3, 5, 3, 4, 5, 4, 5, 6, 5, 2, 7, 1, 4, 2, 3, 1, 6, 3, 4, 7, 3, 3, 3, 5, 5, 4, 3, 8, 3, 6, 2, 4, 3, 4, 5, 6, 6, 5, 5, 4, 5

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

The smallest k! with n prime factors occurs for n in A060250.

103!+1 = 27437*31084943*C153, so a(103) is unknown until this 153-digit composite is factored. a(104) = 4 and a(105) = 6. - Rick L. Shepherd, Jun 10 2003

			a(2)=2 because 4! + 1 = 25 = 5*5

Amiram Eldar, Table of n, a(n) for n = 1..139
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations
Paul Leyland, Factors of n!+1.

Cf. A000040 (prime numbers), A001359 (twin primes).
Cf. A038507, A054988, A054989, A054991, A054992.
Cf. A066856 (number of distinct prime divisors of n!+1), A084846 (mu(n!+1)).

a[q_] := Module[{x, n}, x=FactorInteger[q!+1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
A054990[n_Integer] := PrimeOmega[n! + 1]; Table[A054990[n], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

PARI
```
for(n=1,64,print1(bigomega(n!+1),","))
```

More terms from Robert G. Wilson v, Mar 23 2001

More terms from Rick L. Shepherd, Jun 10 2003

A104368 Number of distinct prime factors of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 4, 4, 2, 1, 4, 2, 2, 3, 2, 1, 4, 3, 3, 1, 3, 4, 5, 4, 5, 4, 3, 3, 4, 7, 2, 5, 4, 4, 2, 4, 3, 4, 2, 3, 7, 4, 2, 4, 2, 3, 2, 4, 3, 4, 5, 3, 4, 4, 2, 1, 2, 4, 6, 4, 3, 3, 4, 4, 6, 6, 6, 5, 3, 6, 4, 5, 3, 3, 5, 3, 4, 3, 4, 7, 6, 4, 3, 4, 2, 3, 3, 2, 2, 5, 2, 3, 3, 6, 4, 3, 3, 2, 3, 2, 6

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A001221, A054988, A066856, A104350, A104359, A104360, A104365, A104366, A104367, A104369, A104370, A104371, A104372.

A104350[n_] := Product[FactorInteger[k][[-1, 1]], {k, 1, n}]; PrimeNu[
Table[A104350[n] + 1, {n, 2, 50}]] (* G. C. Greubel, May 10 2017 *)

a(n) = A001221(A104365(n)).

a(51)-a(76) from Amiram Eldar, Feb 12 2020
a(77)-a(81) from Jinyuan Wang, Apr 02 2020
Terms a(82) onward from Max Alekseyev, Oct 03 2022

A366811 The number of divisors of prime(n)#+1 where p# is the product of all the primes from 2 to p inclusive.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 4, 8, 4, 4, 8, 2, 8, 8, 4, 8, 16, 16, 4, 4, 16, 4, 8, 4, 16, 8, 4, 16, 16, 8, 8, 32, 8, 64, 4, 8, 4, 32, 16, 16, 4, 64, 8, 16, 8, 32, 64, 128, 4, 64, 8, 32, 8, 16, 4, 64, 32, 16, 32, 8, 32, 32, 32, 8, 8, 32, 32, 64, 8, 16, 16, 128, 32, 8, 16

Offset: 0

Sean A. Irvine, Oct 23 2023

nonn

			a(6) = 4 because the divisors of 13#+1 = 30031 are {1, 59, 509, 30031}.

Amiram Eldar, Table of n, a(n) for n = 0..98

Cf. A006862, A000005, A054988, A366808, A366812, A064145.

seq(numtheory[tau](mul(ithprime(k), k=1..n)+1), n=0..30);

Map[DivisorSigma[0, #] &, 1 + FoldList[Times, 1, Prime@ Range@ 19] ] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sigma0(prime(n)#+1) = A000005(A006862(n)).

A366812 The sum of the divisors of prime(n)#+1 where p# is the product of all the primes from 2 to p inclusive.

Original entry on oeis.org

3, 4, 8, 32, 212, 2312, 30600, 544880, 9727992, 223796952, 6500793728, 200560490132, 7461870828048, 309238687200848, 13161101219883792, 615548170519961184, 33465582319854797280, 1930276657976815787040, 117814338226489513454272, 7858321551223903311137992

Offset: 0

Sean A. Irvine, Oct 23 2023

nonn

			a(6) = 30600 because the divisors of 13#+1 = 30031 are {1, 59, 509, 30031}.

Amiram Eldar, Table of n, a(n) for n = 0..98

Cf. A006862, A000230, A054988, A366809, A366811, A366757.

seq(numtheory[sigma](mul(ithprime(k), k=1..n)-1), n=0..30);

Map[DivisorSigma[1, #] &, 1 + FoldList[Times, 1, Prime@ Range@ 19] ] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sigma(prime(n)#+1) = A000230(A006862(n)).

cp's OEIS Frontend

A056160 Sum of A054988 and A054989.

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

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A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

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A054991 Number of prime divisors of n! - 1 (counted with multiplicity).

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A054989 Number of prime divisors of -1 + (product of first n primes).

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A054990 Number of prime divisors of n! + 1 (counted with multiplicity).

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A104368 Number of distinct prime factors of A104365(n) = A104350(n) + 1.

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A366811 The number of divisors of prime(n)#+1 where p# is the product of all the primes from 2 to p inclusive.