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A072480 Shadow transform of factorials A000142. 2
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 (list; graph; refs; listen; history; text; internal format)
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

N. J. A. Sloane, Transforms

FORMULA

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

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.

Sequence in context: A104686 A301428 A104514 * A243920 A181839 A100939

Adjacent sequences:  A072477 A072478 A072479 * A072481 A072482 A072483

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified October 23 12:10 EDT 2018. Contains 316527 sequences. (Running on oeis4.)