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A375636
The number of infinitary divisors of n! that are factorials.
3
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
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)));
KEYWORD
nonn
AUTHOR
Amiram Eldar, Aug 22 2024
STATUS
approved