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A134603 Numbers (excluding primes and powers of primes) such that the square mean of their prime factors is an integer (where the square mean of c and d is sqrt((c^2+d^2)/2)).

%I A134603 #18 Apr 09 2025 15:12:42
%S A134603 119,161,351,378,455,527,595,721,845,918,959,1045,1081,1241,1265,1323,
%T A134603 1375,1547,1615,1792,1855,2047,2145,2175,2345,2457,2645,2665,2737,
%U A134603 3281,3367,3509,3713,3835,3887,3995,4207,4305,4347,4625,4633,4655,4681,5000
%N A134603 Numbers (excluding primes and powers of primes) such that the square mean of their prime factors is an integer (where the square mean of c and d is sqrt((c^2+d^2)/2)).
%C A134603 Numbers included in A134600, but not in A025475. a(1)=119 is the minimal number with this property.
%H A134603 Hieronymus Fischer, <a href="/A134603/b134603.txt">Table of n, a(n) for n = 1..10000</a>
%e A134603 a(2) = 161, since 161 = 7*23 and sqrt((7^2+23^2)/2) = sqrt(289) = 17 is an integer.
%e A134603 a(4) = 378, since 378 = 2*3*3*3*7 and sqrt((2^2+3*3^2+7^2)/5) = sqrt(80/5) = 4 is an integer.
%e A134603 a(28519) = 114445555, since 114445555 = 5*7*41*173*461 and sqrt((5^2+7^2+41^2+173^2+461^2)/5) = sqrt(48841) = 221.
%t A134603 f[{a_,b_}]:=Table[a,b];Select[Range[2,5000],!PrimePowerQ[#]&&IntegerQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* _James C. McMahon_, Apr 09 2025 *)
%Y A134603 Cf. A001597, A025475, A134333, A134344, A134376.
%Y A134603 Cf. A134600, A134605, A134608, A134611, A134617, A134619, A134621.
%K A134603 nonn
%O A134603 1,1
%A A134603 _Hieronymus Fischer_, Nov 11 2007
%E A134603 Minor edits by _Hieronymus Fischer_, Apr 21 2013