A072480 Shadow transform of factorials A000142.

0, 1, 0, 0, 0, 0, 3, 0, 4, 3, 5, 0, 8, 0, 7, 10, 10, 0, 12, 0, 15, 14, 11, 0, 20, 15, 13, 18, 21, 0, 25, 0, 24, 22, 17, 28, 30, 0, 19, 26, 35, 0, 35, 0, 33, 39, 23, 0, 42, 35, 40, 34, 39, 0, 45, 44, 49, 38, 29, 0, 55, 0, 31, 56, 56, 52, 55, 0, 51, 46, 63, 0, 66, 0, 37, 65, 57, 66, 65

Offset: 0

Author

N. J. A. Sloane and Vladeta Jovovic, Aug 02 2002

Keywords

• nonn
• easy

Comments

For n > 1, a(n) is the number of solutions (n,k) of k! = n! (mod n) where 1 <= k < n. - Clark Kimberling, Feb 11 2012

For n > 1, a(n) is the smallest number k such that n divides (n - k)! but not (n - k - 1)!. - Jianing Song, Aug 29 2018

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
• OEIS Wiki, Shadow transform.
• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A000142, A002034, A072458.

Programs

Maple

a:= n-> add(`if`(modp(j!, n)=0, 1, 0), j=0..n-1):
seq(a(n), n=0..120);  # Alois P. Heinz, Sep 16 2019

Mathematica

s[k_] := k!;
f[n_, k_] := If[Mod[s[n] - s[k], n] == 0, 1, 0];
t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
a[n_] := Count[Flatten[t[n]], 1]
Table[a[n], {n, 2, 420}] (* A072480 *)
Flatten[Position[%, 0]]  (* A006093, primes-1 *)
(* Agrees with A072480 for n > 1, from Clark Kimberling, Feb 12 2012 *)

PARI

A002034(n) = if(1==n,n,my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); (k)); \\ From A002034
A072480(n) = if(n<2,n,(n-A002034(n))); \\ Antti Karttunen, Oct 01 2018

Formula

For n > 1, a(n) = n - A002034(n).