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A258456 Product of divisors of n is not a square.

%I A258456 #22 May 31 2025 18:48:03
%S A258456 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 A258456 47,48,49,50,52,53,59,61,63,64,67,68,71,73,75,76,79,80,83,89,92,97,98,
%U A258456 99,100,101,103,107,109,112,113,116,117,121,124,127,131,137
%N A258456 Product of divisors of n is not a square.
%C A258456 Numbers n such that A007955(n) is not a square.
%C A258456 Complement of A048943.
%C A258456 2 is only number n from this sequence such that 1 + Product_{d|n} d is a prime.
%C A258456 If 1 + Product_{d|n} d for n > 2 is a prime p, then Product_{d|n} d is a square (see A258455).
%C A258456 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 A258456 Chai Wah Wu, <a href="/A258456/b258456.txt">Table of n, a(n) for n = 1..10000</a>
%e A258456 9 is in sequence because product of divisors of 9 = 1*3*9 = 27 is not square.
%t A258456 Select[Range@ 137, ! IntegerQ@ Sqrt[Times @@ Divisors@ #] &] (* _Michael De Vlieger_, Jun 02 2015 *)
%o A258456 (Magma) [n: n in [1..200] | not IsSquare(&*(Divisors(n)))];
%o A258456 (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 A258456 (Python)
%o A258456 from gmpy2 import iroot
%o A258456 from sympy import divisor_count
%o A258456 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 A258456 Cf. A007955, A118369, A118370, A258455.
%K A258456 nonn,easy
%O A258456 1,1
%A A258456 _Jaroslav Krizek_, May 30 2015