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A326666 Numbers k such that there exists a factorization of k into factors > 1 whose mean is not an integer but whose geometric mean is an integer.

%I A326666 #13 Jul 02 2021 01:53:59
%S A326666 36,64,100,144,196,216,256,324,400,484,512,576,676,784,900,1000,1024,
%T A326666 1156,1296,1444,1600,1728,1764,1936,2116,2304,2500,2704,2744,2916,
%U A326666 3136,3364,3375,3600,3844,4096,4356,4624,4900,5184,5476,5776,5832,6084,6400,6724
%N A326666 Numbers k such that there exists a factorization of k into factors > 1 whose mean is not an integer but whose geometric mean is an integer.
%H A326666 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A326666 36 has two such factorizations: (3*12) and (4*9).
%e A326666 The sequence of terms together with their prime indices begins:
%e A326666     36: {1,1,2,2}
%e A326666     64: {1,1,1,1,1,1}
%e A326666    100: {1,1,3,3}
%e A326666    144: {1,1,1,1,2,2}
%e A326666    196: {1,1,4,4}
%e A326666    216: {1,1,1,2,2,2}
%e A326666    256: {1,1,1,1,1,1,1,1}
%e A326666    324: {1,1,2,2,2,2}
%e A326666    400: {1,1,1,1,3,3}
%e A326666    484: {1,1,5,5}
%e A326666    512: {1,1,1,1,1,1,1,1,1}
%e A326666    576: {1,1,1,1,1,1,2,2}
%e A326666    676: {1,1,6,6}
%e A326666    784: {1,1,1,1,4,4}
%e A326666    900: {1,1,2,2,3,3}
%e A326666   1000: {1,1,1,3,3,3}
%e A326666   1024: {1,1,1,1,1,1,1,1,1,1}
%e A326666   1156: {1,1,7,7}
%e A326666   1296: {1,1,1,1,2,2,2,2}
%e A326666   1444: {1,1,8,8}
%t A326666 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A326666 Select[Range[1000],Length[Select[facs[#],!IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]]>1&]
%Y A326666 A subsequence of A001597.
%Y A326666 Factorizations with integer mean are A326622.
%Y A326666 Factorizations with integer geometric mean are A326028.
%Y A326666 Cf. A001055, A067538, A067539, A326027, A326516, A326623, A326641, A326643, A326645, A326647.
%K A326666 nonn
%O A326666 1,1
%A A326666 _Gus Wiseman_, Jul 17 2019