A258456 Product of divisors of n is not a square.

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, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 112, 113, 116, 117, 121, 124, 127, 131, 137

Offset: 1

Author

Jaroslav Krizek, May 30 2015

Keywords

• nonn
• easy

Comments

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

Complement of A048943.

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

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

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

Examples

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

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007955, A118369, A118370, A258455.

Programs

Magma

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

Mathematica

Select[Range@ 137, ! IntegerQ@ Sqrt[Times @@ Divisors@ #] &] (* Michael De Vlieger, Jun 02 2015 *)

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

Python

from gmpy2 import iroot
from sympy import divisor_count
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