A069149 Numbers k such that k*k!/A062758(k) is an integer where A062758(k) is the product of squares of divisors of k.

1, 2, 3, 5, 7, 9, 11, 12, 13, 15, 17, 19, 21, 23, 24, 25, 27, 29, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 69, 70, 71, 72, 73, 75, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 90, 91, 93, 95, 96, 97, 98, 99

Offset: 1

Author

Benoit Cloitre, Apr 08 2002

Keywords

• easy
• nonn

Comments

Also numbers k such that k! is divisible by k ^ (tau(k) - 1). - David A. Corneth, Apr 23 2018

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A069148.

Cf. A001563 (n*n!), A062758.

Programs

Mathematica

Select[Range[100], Divisible[#!*#, Times @@ Divisors[#]^2] &] (* Ivan Neretin, Apr 22 2018 *)
Select[Range[100], Divisible[#!, #^(DivisorSigma[0, #] - 1)] &] (* Amiram Eldar, Jul 07 2022 *)

PARI

for(n=1, 320, if((n*(n!))%(n^numdiv(n))==0, print1(n, ", ")))

PARI

is(n) = {my(f = factor(n), qdiv = numdiv(f)); for(i = 1, #f~, cn = n; t = 0; while(cn \= f[i,1], t += cn); if(t < f[i,2] * (qdiv - 1), return(0))); 1} \\ David A. Corneth, Apr 27 2018