A102067 Numbers k such that k does not divide P(k)! even though P(k)^2 is not a factor of k, where P(k) is the largest prime factor of k.

12, 24, 45, 48, 80, 90, 96, 135, 160, 175, 180, 189, 192, 224, 240, 270, 320, 350, 360, 378, 384, 405, 448, 480, 525, 539, 540, 567, 637, 640, 672, 700, 720, 756, 768, 810, 875, 896, 945, 960, 1050, 1078, 1080, 1120, 1134, 1215, 1274, 1280, 1344, 1375, 1400, 1440

Offset: 1

Author

Jonathan Sondow, Dec 28 2004

Keywords

• nonn

Comments

Clearly, if P(k)^2 is a factor of k, then k does not divide P(k)!. Each member shows that the converse is false.

k is a member if and only if k is in A057109 but not in A070003.

Examples

			12 does not divide P(12)! = 3! and 3^2 is not a factor of 12.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Paul Erdős and Ilias Kastanas, Solution 6674: The smallest factorial that is a multiple of n, Amer. Math. Monthly, Vol. 101, No. 2 (1994), p. 179.
• Aubrey J. Kempner, Miscellanea, Amer. Math. Monthly, Vol. 25, No. 5 (1918), pp. 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n."
• Eric Weisstein's World of Mathematics, Greatest Prime Factor.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A006530, A057109, A070003, A102068.

Programs

Mathematica

q[n_] := Module[{p = FactorInteger[n][[-1, 1]]}, !Divisible[n, p^2] && !Divisible[p!, n]]; Select[Range[1500], q] (* Amiram Eldar, Mar 30 2021 *)

PARI

isok(n) = {my(f = factor(n)); my(P = f[#f~,1]); (P! % n) && (n % P^2);} \\ Michel Marcus, Sep 16 2015

Extensions

More terms from Michel Marcus, Sep 16 2015