A054989 -id:A054989 - 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

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

A054988 Number of prime divisors of 1 + (product of first n primes), with multiplicity.

Original entry on oeis.org

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

Offset: 1

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

Prime divisors are counted with multiplicity. - Harvey P. Dale, Oct 23 2020

It is an open question as to whether omega(p#+1) = bigomega(p#+1) = a(n); that is, as to whether the Euclid numbers are squarefree. Any square dividing p#+1 must exceed 2.5*10^15 (see Vardi, p. 87). - Sean A. Irvine, Oct 21 2023

			a(6)=2 because 2*3*5*7*11*13+1 = 30031 = 59 * 509.

Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991.

Max Alekseyev, Table of n, a(n) for n = 1..98
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A001222, A002585, A006862, A014545, A054989, A054990, A054991, A054992.

A054988 := proc(n)
    numtheory[bigomega](1+mul(ithprime(i),i=1..n)) ;
end proc:
seq(A054988(n),n=1..20) ; # R. J. Mathar, Mar 09 2022

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[#+1]&/@FoldList[Times,Prime[Range[90]]] (* Harvey P. Dale, Oct 23 2020 *)

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

a(n) = Omega(1 + Product_{k=1..n} prime(k)). - Wesley Ivan Hurt, Mar 06 2022
a(n) = A001222(A006862(n)). - Michel Marcus, Mar 07 2022
a(n) = 1 iff n is in A014545. - Bernard Schott, Mar 07 2022

More terms from Robert G. Wilson v, Mar 24 2001
a(44)-a(81) from Charles R Greathouse IV, May 07 2011
a(82)-a(87) from Amiram Eldar, Oct 03 2019

A104360 Number of distinct prime factors of A104357(n) = A104350(n) - 1.

Original entry on oeis.org

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

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 2..145
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A001221, A054989, A066877, A104350, A104357, A104358, A104359, A104361, A104362, A104363, A104364, A104368.

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(A104357(n)).

a(51)-a(74) from Amiram Eldar, Feb 12 2020
More terms from Jinyuan Wang, Apr 02 2020
Terms a(90) onward from Max Alekseyev, Oct 03 2022

A059958 Smallest number m such that m*(m+1) has at least n distinct prime factors.

Original entry on oeis.org

1, 2, 5, 14, 65, 209, 714, 7314, 38570, 254540, 728364, 11243154, 58524465, 812646120, 5163068910, 58720148850, 555409903685, 4339149420605, 69322940121435, 490005293940084, 5819629108725509, 76622240600506314

Offset: 1

Labos Elemer, Mar 02 2001

The original definition left unclear whether "at least" or "exactly" n prime factors are required. Now the "at least" variant was chosen, for the other variant ("exactly"), see A069354: At least up to a(18), both criteria yield the same number, and therefore a(n) = A069354(n) - 1, since m and m+1 are always coprime. - M. F. Hasler, Jan 15 2014
10^13 < a(19) <= 69322940121435. - Giovanni Resta, Mar 24 2020
Terms a(1)-a(10) appear in Erdős and Nicolas (1978-1979). - Amiram Eldar, Jun 24 2023

			For n = 9, a(9)*(a(9) + 1) = 38570*38571 = (2*5*7*19*29)*(3*13*23*43) with 9 distinct prime factors.

Michael S. Branicky, Python program.
Paul Erdős and Jean-Louis Nicolas, Sur la fonction "nombre de facteurs premiers de n", Séminaire Delange-Pisot-Poitou, Théorie des nombres, Vol. 20, No. 2 (1978-1979), Talk no. 32, pp. 1-19. See p. 10.

Cf. A001221, A002110, A006549, A054989, A083002, A232096, A232097.

With[{s = Map[PrimeNu[Times @@ #] &, Partition[Range[10^6], 2, 1]]}, Array[FirstPosition[s, n_/; n>=#][[1]] &, Max@ s]] (* Michael De Vlieger, Nov 02 2017 *)

a(n) = my(m=1); while(omega(m*(m+1)) < n, m++); m; \\ Michel Marcus, Jul 09 2018

a(n) = Min_{ m | A001221(m*(m+1)) >= n }.
a(n) <= A002110(n) - 1 because A001221((q-1)*q) >= n+1 for q = A002110(n).
Conjecture: a(n) = A069354(n) - 1. - Robert G. Wilson v, Feb 18 2014

More terms from William Rex Marshall, Mar 18 2001
Offset corrected and a(15)-a(16) from Donovan Johnson, Jan 31 2009
a(17) from Donovan Johnson, Sep 15 2010
a(18) from Don Reble, Jan 15 2014
Edited by M. F. Hasler, Jan 15 2014
a(19)-a(20) from Michael S. Branicky, Feb 08 2023
a(21) from Michael S. Branicky, Feb 10 2023
a(22) from Michael S. Branicky, Feb 23 2023

A366809 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

1, 6, 30, 240, 2310, 30030, 518940, 9943560, 230876448, 6551588160, 200561595684, 7471933410000, 304250263527210, 13082853940673340, 618109122639794688, 32589631537463089128, 1922760350251477679196, 117386696543681561301312, 7906535060701218163040640

Offset: 1

Sean A. Irvine, Oct 23 2023

nonn

			a(4)=240 because the divisors of 7#-1 = 209 are {1, 11, 19, 209}.

Cf. A057588, A000230, A054989, A366809, A064145.

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

a(n) = sigma(prime(n)#-1) = A000230(A057588(n)).

A366808 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

1, 2, 2, 4, 2, 2, 4, 8, 8, 4, 4, 16, 2, 4, 8, 8, 4, 8, 8, 4, 4, 16, 8, 2, 4, 4, 16, 16, 16, 8, 8, 8, 8, 8, 8, 16, 16, 16, 32, 8, 32, 16, 32, 16, 16, 16, 16, 4, 8, 8, 4, 16, 8, 16, 4, 16, 16, 128, 16, 8, 8, 16, 16, 8, 8, 2, 8, 2, 16, 8, 32, 32, 16, 16, 64, 32

Offset: 1

Sean A. Irvine, Oct 23 2023

nonn

			a(4)=4 because the divisors of 7#-1 = 209 are {1, 11, 19, 209}.

Cf. A057588, A000005, A054989, A366809, A064145.

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

Map[DivisorSigma[0, #] &, -1 + FoldList[Times, Prime@ Range@ 30] ] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sigma0(prime(n)#-1) = A000005(A057588(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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A054990 Number of prime divisors of n! + 1 (counted with multiplicity).

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Mathematica

PARI

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A054988 Number of prime divisors of 1 + (product of first n primes), with multiplicity.

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A104360 Number of distinct prime factors of A104357(n) = A104350(n) - 1.

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