A345935 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!.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 4, 1, 3, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 2, 1, 6, 1, 2, 3, 2, 2, 4, 1, 3, 2, 4, 1, 4, 1, 2, 2, 3, 2, 4, 1, 2, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 2, 1, 2, 3, 3, 1, 4, 1, 4, 4

Offset: 1

Author

Antti Karttunen, Jul 02 2021

Keywords

• nonn

Examples

			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 three divisors that obtain the maximal value 6 obtained at 36 itself, therefore a(36) = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000005, A002034, A345934, A345936, A345944 (positions of 1's), A345945 (of terms > 1), A345950.

Cf. also A344590.

Programs

Mathematica

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 *)

PARI

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.
A345935(n) = { my(x=A002034(n)); sumdiv(n,d,A002034(d)==x); };

Formula

a(n) = Sum_{d|n} [A002034(d) = A002034(n)], where [ ] is the Iverson bracket.

a(n) = A000005(n) - A345936(n).

a(n) <= A345934(n).