A054992 -id:A054992 - cp's OEIS frontend

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

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

A057938 Number of prime factors of 6^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 4, 2, 3, 3, 2, 2, 4, 3, 5, 3, 3, 6, 3, 3, 6, 4, 3, 4, 4, 4, 5, 5, 4, 9, 2, 3, 6, 3, 11, 5, 4, 3, 9, 5, 4, 7, 2, 3, 7, 5, 2, 7, 7, 6, 8, 4, 5, 10, 8, 6, 7, 3, 2, 6, 3, 2, 10, 3, 8, 11, 5, 5, 6, 7, 4, 5, 6, 5, 10, 5, 6, 11, 6, 6, 8, 4, 5, 8, 8, 5, 11, 4, 3, 16, 12, 4, 5, 5, 10, 4, 4, 5

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..430 (first 408 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), this sequence (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A057955, A062394, A274904.

f:=func; [f(6^n + 1):n in [1..100]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[6^Range[100]+1] (* Harvey P. Dale, Mar 10 2013 *)

a(n) = A057955(2n) - A057955(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A062394(n)). - Amiram Eldar, Feb 02 2020

A069061 Sum of divisors of 2^n+1.

Original entry on oeis.org

4, 6, 13, 18, 48, 84, 176, 258, 800, 1302, 2736, 4356, 10928, 20520, 51792, 65538, 174768, 351120, 699056, 1110276, 3100240, 5048232, 11184816, 17041416, 49012992, 82623888, 211053040, 284225796, 727960800, 1494039792, 2863311536, 4301668356, 12611914848, 20788904016

Offset: 1

Benoit Cloitre, Apr 04 2002

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1128 (terms 1..1062 from Amiram Eldar)

Cf. A000051, A000203, A002185, A002587, A002589, A046798, A046799, A053285, A054992, A057957, A059886, A085029, A086257, A112092, A250291, A274906, A295501.

Row sums of A374237.

DivisorSigma[1, 2^Range[50] + 1] (* Paolo Xausa, Jul 05 2024 *)

a(n) = sigma(2^n+1); \\ Michel Marcus, Nov 24 2013

a(n) = sigma(2^n+1).

a(n) = A000203(A000051(n)). - Michel Marcus, Nov 24 2013

More terms from Amiram Eldar, Oct 04 2019

A053285 Totient of 2^n+1.

Original entry on oeis.org

1, 2, 4, 6, 16, 20, 48, 84, 256, 324, 800, 1364, 3840, 5460, 12544, 19800, 65536, 87380, 186624, 349524, 986880, 1365336, 3345408, 5592404, 16515072, 20250000, 52306176, 84768120, 252645120, 351847488, 760320000, 1431655764, 4288266240, 5632621632, 13628740608

Offset: 0

Labos Elemer, Mar 03 2000

nonn

			It is a power of 2 iff n is a Fermat prime.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..300 from Robert Israel; terms 301..1062 from Amiram Eldar)

Cf. A000010, A000225, A002587, A000051, A046798, A046799, A049479, A051953, A054992.

[EulerPhi(2^n+1) : n in [1..40]]; // Vincenzo Librandi, Aug 12 2015

seq(numtheory:-phi(2^n+1), n=0..50); # Robert Israel, Aug 12 2015

Table[EulerPhi[2^n + 1], {n, 35}] (* Vincenzo Librandi, Aug 12 2015 *)

vector(40, n, eulerphi(2^n+1)) \\ Michel Marcus, Aug 12 2015

a(n) = A000010(A000051(n)).

a(0)=1 prepended by Alois P. Heinz, Aug 12 2015

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

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

A193330 Number of prime factors of n^2 + 1, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 2, 3, 1, 2, 1, 3, 3, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 2, 2, 3, 3, 3, 2, 2, 1, 3, 3, 2, 1, 3, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 1, 3, 1, 5, 2, 2, 2, 2, 2, 3, 2, 2, 1, 3, 4, 2, 3, 2, 3, 4, 1, 3, 2, 3, 2, 2, 2, 3, 3, 4, 1, 2

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 22 2011

nonn

a(n) is also the number of terms, counted with multiplicity, in a prime factorization of n + i in the ring of Gaussian integers. - Jason Kimberley, Dec 17 2011

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A046051, A054992, A193295.

A193330[n_Integer] := PrimeOmega[n^2 + 1];A193330 /@ Range[0, 100]
PrimeOmega/@(Range[0,90]^2+1) (* Harvey P. Dale, Apr 03 2019 *)

a(n)=bigomega(n^2+1) \\ Charles R Greathouse IV, Jul 23 2011

A366713 Number of prime factors of 12^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 3, 3, 3, 5, 4, 2, 4, 5, 4, 7, 2, 3, 5, 4, 6, 9, 6, 4, 6, 6, 5, 10, 3, 3, 6, 3, 2, 9, 4, 5, 7, 4, 5, 11, 4, 5, 6, 5, 4, 12, 3, 5, 5, 5, 10, 9, 5, 5, 10, 7, 9, 11, 8, 6, 10, 5, 6, 15, 5, 9, 11, 4, 5, 12, 10, 3, 10, 5, 8, 17, 5, 6, 9, 4, 6, 15

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A001222, A054992, A057934, A057935, A057936, A057937, A057938, A057939, A057940, A057941, A366712, A366714, A366715, A366716, A366719, A366720, A366708, A366687.

Mathematica
```
PrimeOmega[12^Range[70]+1]
```
PARI
```
a(n)=bigomega(12^n+1)
```

a(n) = bigomega(12^n+1) = A001222(A178248(n)).

A086257 Number of primitive prime factors of 2^n+1.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3, 1, 1, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 1, 1, 2, 2, 1, 2, 1, 4, 2, 2, 1, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 1, 4, 1, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 4, 1, 3, 3, 4, 1, 2, 3, 4, 5, 2, 1, 4, 1, 3, 3, 3, 3, 1, 2, 3, 2, 1, 4, 3, 2, 4, 1, 4, 2, 1

Offset: 0

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 rA086258 for those n that have a record number of primitive prime factors.

			a(14) = 2 because 2^14+1 = 5*29*113 and 29 and 113 do not divide 2^r+1 for r < 14.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..500 from T. D. Noe; terms 501..1062 from Amiram Eldar)
Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Excluding a(0) = 1, forms a bisection of A086251.

Cf. A046799 (number of distinct prime factors of 2^n+1), A054992 (number of prime factors, with repetition, of 2^n+1), A086258.

nMax=200; pLst={}; Table[f=Transpose[FactorInteger[2^n+1]][[1]]; f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 0, nMax}]

For n > 0, a(n) = A086251(2*n). - Max Alekseyev, Sep 06 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A057938 Number of prime factors of 6^n + 1 (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A069061 Sum of divisors of 2^n+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A053285 Totient of 2^n+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple