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

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to factorial numbers

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).

Example

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.

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 *)

Prog

(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); };

Crossrefs

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

Cf. also A344590.

Sequence in context: A371090 A064547 A318306 * A214715 A244145 A371734

Adjacent sequences: A345932 A345933 A345934 * A345936 A345937 A345938

Keyword

nonn

Author

Antti Karttunen, Jul 02 2021

Status

approved