A134604 Numbers (excluding primes and powers of primes) such that the square mean of their prime factors is a prime (where the square mean of c and d is sqrt((c^2+d^2)/2)).

119, 161, 351, 595, 721, 845, 959, 1045, 1081, 1241, 1323, 1375, 1547, 1792, 1855, 2457, 2645, 2737, 3281, 3367, 3509, 3887, 3995, 4347, 4625, 4655, 4681, 5376, 5795, 6545, 6615, 6643, 6993, 7505, 7705, 7803, 7889, 8019, 9295, 9625, 10557, 11845

Offset

1,1

Comments

Numbers included in A134601, but not in A025475. a(1)=119 is the minimal number with this property.

Links

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

Example

a(2) = 161, since 161 = 7*23 and sqrt((7^2+23^2)/2) = sqrt(289)=17 is a prime.
a(10183) = 114383711 = 13*83*227*467 and sqrt((13^2+83^2+227^2+467^2)/4) = sqrt(69169) = 263 is a prime.

Mathematica

f[{a_, b_}]:=Table[a, b]; Select[Range[2, 11845], !PrimePowerQ[#]&&PrimeQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 09 2025 *)

Crossrefs

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

Cf. A134600, A134602, A134608, A134611, A134617, A134619, A134621.

Sequence in context: A273179 A247137 A134603 * A255580 A227515 A207059

Adjacent sequences: A134601 A134602 A134603 * A134605 A134606 A134607

Keyword

nonn

Author

Hieronymus Fischer, Nov 11 2007

Extensions

Minor edits by Hieronymus Fischer, Apr 22 2013

Status

approved