A375637 - OEIS

A375637

Positive numbers k such that k! does not have nontrivial infinitary divisors that are factorials.

1, 2, 6, 10, 18, 20, 24, 30, 34, 35, 36, 46, 48, 49, 54, 66, 68, 69, 72, 78, 81, 86, 87, 90, 91, 92, 96, 102, 108, 114, 116, 117, 120, 121, 126, 130, 142, 143, 150, 155, 156, 161, 166, 171, 172, 180, 184, 190, 192, 198, 204, 205, 212, 216, 222, 228, 232, 238, 240

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OFFSET

1,2

COMMENTS

The trivial infinitary divisors of a number m are 1 and m itself. Therefore if k >=2 then k! has at least 2 infinitary divisors that are factorials, 1 and k!.

Numbers k such that A375636(k) <= 2, or equivalently, 1 and numbers k such that A375636(k) = 2.

LINKS

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

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]]; q[n_] := Sum[expQ[e[n], e[m]], {m, 2, n}] <= 1; Select[Range[240], q]

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];

is(n) = sum(m = 2, n, isexp(e(n), e(m))) <= 1;

CROSSREFS

Cf. A000142, A037445, A077609, A375635, A375636.

Sequence in context: A274679 A229218 A255174 * A140777 A309502 A281664

Adjacent sequences: A375634 A375635 A375636 * A375638 A375639 A375640

KEYWORD

nonn

AUTHOR

Amiram Eldar, Aug 22 2024

STATUS

approved