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

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, 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, 2, 3, 4, 2, 3, 3, 6, 4, 4, 2, 3, 4, 2, 3, 4, 4, 4, 2, 2

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) >= 2 for n >= 2.

a(n) <= 2 if and only if n is in A375637.

a(A375638(n)) = n or -1.

a(p) = a(p-1) + 1 for a prime p.

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.

Mathematica

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];
e[n_] := e[n] = FactorInteger[n!][[;; , 2]]; a[n_] := 1 + Sum[expQ[e[n], e[m]], {m, 2, n}]; Array[a, 100]

Prog

(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; }
e(n) = factor(n!)[, 2];
a(n) = 1 + sum(m = 2, n, isexp(e(n), e(m)));

Crossrefs

Cf. A000142, A000720, A037445, A077609, A090971, A375635, A375637, A375638.

Sequence in context: A094937 A215089 A329243 * A161768 A213925 A141810

Adjacent sequences: A375633 A375634 A375635 * A375637 A375638 A375639

Keyword

nonn

Author

Amiram Eldar, Aug 22 2024

Status

approved