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

Author

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

Keywords

• nonn
• hard

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

Examples

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

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

Crossrefs

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

Programs

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 *)

PARI

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

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

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