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%I #48 Oct 21 2023 16:38:23
%S 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,
%T 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,
%U 4,4,7,5,3,4,1,2,5,5,5,4,5,3,5,4,6,5,8
%N Number of prime divisors of 1 + (product of first n primes), with multiplicity.
%C Prime divisors are counted with multiplicity. - _Harvey P. Dale_, Oct 23 2020
%C 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
%D Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991.
%H Max Alekseyev, <a href="/A054988/b054988.txt">Table of n, a(n) for n = 1..98</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha102.htm">Factorizations of many number sequences</a>
%H R. G. Wilson v, <a href="/A038507/a038507.txt">Explicit factorizations</a>
%F a(n) = Omega(1 + Product_{k=1..n} prime(k)). - _Wesley Ivan Hurt_, Mar 06 2022
%F a(n) = A001222(A006862(n)). - _Michel Marcus_, Mar 07 2022
%F a(n) = 1 iff n is in A014545. - _Bernard Schott_, Mar 07 2022
%e a(6)=2 because 2*3*5*7*11*13+1 = 30031 = 59 * 509.
%p A054988 := proc(n)
%p numtheory[bigomega](1+mul(ithprime(i),i=1..n)) ;
%p end proc:
%p seq(A054988(n),n=1..20) ; # _R. J. Mathar_, Mar 09 2022
%t 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}]]
%t PrimeOmega[#+1]&/@FoldList[Times,Prime[Range[90]]] (* _Harvey P. Dale_, Oct 23 2020 *)
%o (PARI) a(n) = bigomega(1+prod(k=1, n, prime(k))); \\ _Michel Marcus_, Mar 07 2022
%Y Cf. A001222, A002585, A006862, A014545, A054989, A054990, A054991, A054992.
%K nonn,hard
%O 1,6
%A Arne Ring (arne.ring(AT)epost.de), May 30 2000
%E More terms from _Robert G. Wilson v_, Mar 24 2001
%E a(44)-a(81) from _Charles R Greathouse IV_, May 07 2011
%E a(82)-a(87) from _Amiram Eldar_, Oct 03 2019
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