A375960 Numbers whose product of proper divisors is a cube.

1, 2, 3, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 30, 31, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 88, 89, 97, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 125, 127, 128, 130, 131, 135, 136, 137, 138, 139, 149

Offset: 1

Author

Stefano Spezia, Sep 04 2024

Keywords

• nonn
• easy

Comments

Wells erroneously writes that the smallest number on this list should be 24.

All the primes are in this list since they have the only proper divisor 1 which is trivially a cube.

Examples

			16 is a term since 1*2*4*8 = 64 = 4^3.
		

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 101.

Links

• Stefano Spezia, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040 (subsequence), A000578, A007956, A027751, A032741.

Complement of A375962.

Programs

Mathematica

Select[Range[150],IntegerQ[Product[Part[Divisors[#],i],{i,DivisorSigma[0,#]-1}]^(1/3)] &]

PARI

isok(k) = my(d=divisors(k)); ispower(vecprod(Vec(d, #d-1)), 3); \\ Michel Marcus, Sep 04 2024