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A345936 Number of divisors d of n for which A002034(d) < A002034(n), where A002034(n) is the smallest positive integer k such that n divides k!.

%I A345936 #10 Jul 04 2021 18:25:49
%S A345936 0,1,1,2,1,2,1,2,2,2,1,4,1,2,2,4,1,4,1,3,2,2,1,4,2,2,3,3,1,4,1,5,2,2,
%T A345936 2,6,1,2,2,4,1,4,1,3,4,2,1,8,2,4,2,3,1,6,2,4,2,2,1,6,1,2,3,5,2,4,1,3,
%U A345936 2,4,1,8,1,2,4,3,2,4,1,8,3,2,1,6,2,2,2,4,1,8,2,3,2,2,2,10,1,4,3,6,1,4,1,4,4
%N A345936 Number of divisors d of n for which A002034(d) < A002034(n), where A002034(n) is the smallest positive integer k such that n divides k!.
%H A345936 Antti Karttunen, <a href="/A345936/b345936.txt">Table of n, a(n) for n = 1..65537</a>
%H A345936 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A345936 a(n) = Sum_{d|n} [A002034(d) < A002034(n)], where [ ] is the Iverson bracket.
%F A345936 a(n) = A000005(n) - A345935(n).
%e A345936 36 has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36. When A002034 is applied to them, one obtains values [1, 2, 3, 4, 3, 6, 4, 6, 6], thus there are six divisors that do not obtain the maximal value 6 obtained at 36 itself, therefore a(36) = 6.
%t A345936 a[n_]:=(m=1;While[Mod[m!,n]!=0,m++];m);Table[Length@Select[Divisors@k,a@#<a@k&],{k,100}] (* _Giorgos Kalogeropoulos_, Jul 03 2021 *)
%o A345936 (PARI)
%o A345936 A002034(n) = if(1==n,n,my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); (k)); \\ After code in A002034.
%o A345936 A345936(n) = { my(x=A002034(n)); sumdiv(n,d,A002034(d)<x); };
%Y A345936 Cf. A000005, A002034, A345935.
%Y A345936 Cf. also A344589.
%K A345936 nonn
%O A345936 1,4
%A A345936 _Antti Karttunen_, Jul 02 2021