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

0,7

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.

Formula

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

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 *)

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)); \\ From A002034
A072480(n) = if(n<2, n, (n-A002034(n))); \\ Antti Karttunen, Oct 01 2018

Crossrefs

Cf. A000142, A002034, A072458.

Sequence in context: A301428 A104514 A349915 * A243920 A344905 A181839

Adjacent sequences: A072477 A072478 A072479 * A072481 A072482 A072483

Keyword

nonn,easy

Author

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

Status

approved