cp's OEIS Frontend

A134600 Composite numbers such that the square mean of their prime factors is an integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

%I A134600 #14 Apr 08 2025 13:09:04
%S A134600 4,8,9,16,25,27,32,49,64,81,119,121,125,128,161,169,243,256,289,343,
%T A134600 351,361,378,455,512,527,529,595,625,721,729,841,845,918,959,961,1024,
%U A134600 1045,1081,1241,1265,1323,1331,1369,1375,1547,1615,1681,1792,1849,1855
%N A134600 Composite numbers such that the square mean of their prime factors is an integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).
%C A134600 All perfect prime powers (A025475) with power > 0 are included.
%C A134600 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
%H A134600 Hieronymus Fischer, <a href="/A134600/b134600.txt">Table of n, a(n) for n = 1..10000</a>
%e A134600 a(5) = 25, since 25=5*5 and sqrt((5^2+5^2)/2)=5;
%e A134600 a(23) = 378, since 378=2*3*3*3*7 and sqrt((2^2+3*3^2+7^2)/5)=sqrt(16)=4.
%t A134600 f[{a_,b_}]:=Table[a,b];Select[Range[2,1855],!PrimeQ[#]&&IntegerQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* _James C. McMahon_, Apr 08 2025 *)
%Y A134600 Cf. A001597, A025475, A134333, A134344, A134376.
%Y A134600 Cf. A134601, A134605, A134608, A134611, A134617, A134619, A134621.
%K A134600 nonn
%O A134600 1,1
%A A134600 _Hieronymus Fischer_, Nov 11 2007
%E A134600 Definition clarified and edited by _Hieronymus Fischer_, Apr 20 2013