login
A375637
Positive numbers k such that k! does not have nontrivial infinitary divisors that are factorials.
2
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
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Aug 22 2024
STATUS
approved