A134612 Nonprime numbers such that the root mean cube of their prime factors is a prime (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921

Offset: 1

Author

Hieronymus Fischer, Nov 11 2007

Keywords

• nonn

Comments

The prime factors are taken with multiplicity.

All perfect prime powers (A025475) with power > 1 are included. First term not included in A025475 is a(211) = 707265 = A134614(5) = A134615(1).

Originally, the first term was 1. This was wrong, since the cube mean of the prime factors of 1 is zero, by definition of the empty sum.

Examples

			a(5) = 25, since 25 = 5*5 and ((5^3+5^3)/2)^(1/3) = 5.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..8600

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134614, A134617, A134619, A134621.

Programs

PARI

lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && ispower(s, 3, &p) && isprime(p), print1(i, ", "));););}  \\ Michel Marcus, Apr 14 2013

Extensions

Edited by Hieronymus Fischer, May 30 2013