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A375636 The number of infinitary divisors of n! that are factorials.

%I A375636 #9 Sep 05 2024 16:23:28
%S A375636 1,2,3,4,5,2,3,5,3,2,3,4,5,5,3,5,6,2,3,2,3,5,6,2,3,3,5,4,5,2,3,6,12,2,
%T A375636 2,2,3,5,3,4,5,7,8,4,4,2,3,2,2,3,6,4,5,2,3,5,7,4,5,4,5,5,3,4,12,2,3,2,
%U A375636 2,3,4,2,3,3,6,4,4,2,3,4,2,3,4,4,4,2,2
%N A375636 The number of infinitary divisors of n! that are factorials.
%H A375636 Amiram Eldar, <a href="/A375636/b375636.txt">Table of n, a(n) for n = 1..10000</a>
%F A375636 a(n) >= 2 for n >= 2.
%F A375636 a(n) <= 2 if and only if n is in A375637.
%F A375636 a(A375638(n)) = n or -1.
%F A375636 a(p) = a(p-1) + 1 for a prime p.
%F A375636 a(n) = 1 + Sum_{k=2..n} [Sum_{p prime <= A007917(k)} A090971(v_p(n!), v_p(k!)) = primepi(k)], where v_p(n) is the p-adic valuation of n, primepi(k) = A000720(k), and [] is the Iverson bracket.
%t A375636 expQ[e1_, e2_] := Module[{m = Length[e2], ans = 1}, Do[If[BitAnd[e1[[i]], e2[[i]]] < e2[[i]], ans = 0; Break[]], {i, 1, m}]; ans];
%t A375636 e[n_] := e[n] = FactorInteger[n!][[;; , 2]]; a[n_] := 1 + Sum[expQ[e[n], e[m]], {m, 2, n}]; Array[a, 100]
%o A375636 (PARI) isexp(e1, e2) = {my(m = #e2, ans = 1); for(i=1,m,if(bitand(e1[i], e2[i]) < e2[i], ans = 0; break)); ans;}
%o A375636 e(n) = factor(n!)[,2];
%o A375636 a(n) = 1 + sum(m = 2, n, isexp(e(n), e(m)));
%Y A375636 Cf. A000142, A000720, A037445, A077609, A090971, A375635, A375637, A375638.
%K A375636 nonn
%O A375636 1,2
%A A375636 _Amiram Eldar_, Aug 22 2024