A134601 Composite numbers such that the square mean of their prime factors is a prime (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 119, 121, 125, 128, 161, 169, 243, 256, 289, 343, 351, 361, 512, 529, 595, 625, 721, 729, 841, 845, 959, 961, 1024, 1045, 1081, 1241, 1323, 1331, 1369, 1375, 1547, 1681, 1792, 1849, 1855, 2048, 2187, 2197, 2209, 2401

Offset: 1

Author

Hieronymus Fischer, Nov 11 2007

Keywords

• nonn

Comments

All perfect prime powers (A025475) with power > 0 are included.

Originally, the definition started with "Nonprime numbers ..." and the first term was equal to 1. This is misleading, since 1 has no prime factors. - Hieronymus Fischer, Apr 20 2013

Examples

			a(5) = 25, since 25=5*5 and sqrt((5^2+5^2)/2)=5;
a(13) = 119, since 119=7*17 and sqrt((7^2+17^2)/2)=sqrt(169)=13.
		

Links

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

Crossrefs

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

Cf. A134600, A134605, A134608, A134611, A134617, A134619, A134621.

Programs

Mathematica

f[{a_,b_}]:=Table[a,b];Select[Range[2,2401],!PrimeQ[#]&&PrimeQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 08 2025 *)

Extensions

Definition clarified and edited by Hieronymus Fischer, Apr 20 2013