A258456 - OEIS

%I #19 Sep 08 2022 08:46:12

%S 2,3,4,5,7,9,11,12,13,17,18,19,20,23,25,28,29,31,32,36,37,41,43,44,45,

%T 47,48,49,50,52,53,59,61,63,64,67,68,71,73,75,76,79,80,83,89,92,97,98,

%U 99,100,101,103,107,109,112,113,116,117,121,124,127,131,137

%N Product of divisors of n is not a square.

%C Numbers n such that A007955(n) is not a square.

%C Complement of A048943.

%C 2 is only number n from this sequence such that 1 + Product_{d|n} d is a prime.

%C If 1 + Product_{d|n} d for n > 2 is a prime p, then Product_{d|n} d is a square (see A258455).

%C m is a term if and only if m is not a fourth power and the number of divisors of m is not a multiple of 4. - _Chai Wah Wu_, Mar 09 2016

%H Chai Wah Wu, <a href="/A258456/b258456.txt">Table of n, a(n) for n = 1..10000</a>

%e 9 is in sequence because product of divisors of 9 = 1*3*9 = 27 is not square.

%t Select[Range@ 137, ! IntegerQ@ Sqrt[Times @@ Divisors@ #] &] (* _Michael De Vlieger_, Jun 02 2015 *)

%o (Magma) [n: n in [1..200] | not IsSquare(&*(Divisors(n)))]

%o (PARI) for(n=1,100,d=divisors(n);p=prod(i=1,#d,d[i]);if(!issquare(p),print1(n,", "))) \\ _Derek Orr_, Jun 12 2015

%o (Python)

%o from gmpy2 import iroot

%o from sympy import divisor_count

%o A258456_list = [i for i in range(1,10**3) if not iroot(i,4)[1] and divisor_count(i) % 4] # _Chai Wah Wu_, Mar 10 2016

%Y Cf. A007955, A118369, A118370, A258455.

%K nonn,easy

%O 1,1

%A _Jaroslav Krizek_, May 30 2015