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!.

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, 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, 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

Offset

1,4

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) - A345935(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 six divisors that do not obtain the maximal value 6 obtained at 36 itself, therefore a(36) = 6.

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.
A345936(n) = { my(x=A002034(n)); sumdiv(n, d, A002034(d)<x); };

Crossrefs

Cf. A000005, A002034, A345935.

Cf. also A344589.

Sequence in context: A223135 A084113 A323086 * A370814 A347050 A348383

Adjacent sequences: A345933 A345934 A345935 * A345937 A345938 A345939

Keyword

nonn

Author

Antti Karttunen, Jul 02 2021

Status

approved