cp's OEIS Frontend

A375637 Positive numbers k such that k! does not have nontrivial infinitary divisors that are factorials.

%I A375637 #7 Aug 22 2024 09:17:54
%S A375637 1,2,6,10,18,20,24,30,34,35,36,46,48,49,54,66,68,69,72,78,81,86,87,90,
%T A375637 91,92,96,102,108,114,116,117,120,121,126,130,142,143,150,155,156,161,
%U A375637 166,171,172,180,184,190,192,198,204,205,212,216,222,228,232,238,240
%N A375637 Positive numbers k such that k! does not have nontrivial infinitary divisors that are factorials.
%C A375637 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!.
%C A375637 Numbers k such that A375636(k) <= 2, or equivalently, 1 and numbers k such that A375636(k) = 2.
%H A375637 Amiram Eldar, <a href="/A375637/b375637.txt">Table of n, a(n) for n = 1..10000</a>
%t A375637 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 A375637 e[n_] := e[n] = FactorInteger[n!][[;; , 2]]; q[n_] := Sum[expQ[e[n], e[m]], {m, 2, n}] <= 1; Select[Range[240], q]
%o A375637 (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 A375637 e(n) = factor(n!)[,2];
%o A375637 is(n) = sum(m = 2, n, isexp(e(n), e(m))) <= 1;
%Y A375637 Cf. A000142, A037445, A077609, A375635, A375636.
%K A375637 nonn
%O A375637 1,2
%A A375637 _Amiram Eldar_, Aug 22 2024