

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.


3



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
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OFFSET

1,1


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.


LINKS

Aubrey J. Kempner, Miscellanea, Amer. Math. Monthly, Vol. 25, No. 5 (1918), pp. 201210. See Section II, "Concerning the smallest integer m! divisible by a given integer n."


EXAMPLE

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


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 *)


PROG

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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



