A134619 Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime.

20, 44, 188, 297, 336, 400, 425, 540, 575, 605, 704, 752, 764, 908, 912, 1025, 1053, 1124, 1172, 1183, 1365, 1380, 1412, 1420, 1452, 1475, 1484, 1519, 1604, 1625, 1809, 1844, 1856, 1936, 1953, 2107, 2192, 2205, 2255, 2320, 2325, 2348, 2368, 2372, 2468

Offset: 1

Author

Hieronymus Fischer, Nov 11 2007

Keywords

• nonn

Examples

			a(10)=605, since 605=5*11*11 and (5^3+11^3+11^3)/3=929 which is prime.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Crossrefs

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

Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134621.

Programs

Mathematica

amcpfQ[n_]:=PrimeQ[Mean[Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]^3]]; Select[ Range[ 2500],amcpfQ] (* Harvey P. Dale, Jun 06 2023 *)

PARI

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

Extensions

Minor edits by Hieronymus Fischer, May 06 2013