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

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

1,6

Comments

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

References

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

Links

Max Alekseyev, Table of n, a(n) for n = 1..98

Hisanori Mishima, Factorizations of many number sequences

R. G. Wilson v, Explicit factorizations

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A340703 A128049 A104543 * A143393 A269111 A166497

Adjacent sequences: A054985 A054986 A054987 * A054989 A054990 A054991

Keyword

nonn,hard

Author

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

Extensions

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

Status

approved