A326666 - OEIS

%I #13 Jul 02 2021 01:53:59

%S 36,64,100,144,196,216,256,324,400,484,512,576,676,784,900,1000,1024,

%T 1156,1296,1444,1600,1728,1764,1936,2116,2304,2500,2704,2744,2916,

%U 3136,3364,3375,3600,3844,4096,4356,4624,4900,5184,5476,5776,5832,6084,6400,6724

%N 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>

%e 36 has two such factorizations: (3*12) and (4*9).

%e The sequence of terms together with their prime indices begins:

%e     36: {1,1,2,2}

%e     64: {1,1,1,1,1,1}

%e    100: {1,1,3,3}

%e    144: {1,1,1,1,2,2}

%e    196: {1,1,4,4}

%e    216: {1,1,1,2,2,2}

%e    256: {1,1,1,1,1,1,1,1}

%e    324: {1,1,2,2,2,2}

%e    400: {1,1,1,1,3,3}

%e    484: {1,1,5,5}

%e    512: {1,1,1,1,1,1,1,1,1}

%e    576: {1,1,1,1,1,1,2,2}

%e    676: {1,1,6,6}

%e    784: {1,1,1,1,4,4}

%e    900: {1,1,2,2,3,3}

%e   1000: {1,1,1,3,3,3}

%e   1024: {1,1,1,1,1,1,1,1,1,1}

%e   1156: {1,1,7,7}

%e   1296: {1,1,1,1,2,2,2,2}

%e   1444: {1,1,8,8}

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Select[Range[1000],Length[Select[facs[#],!IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]]>1&]

%Y A subsequence of A001597.

%Y Factorizations with integer mean are A326622.

%Y Factorizations with integer geometric mean are A326028.

%Y Cf. A001055, A067538, A067539, A326027, A326516, A326623, A326641, A326643, A326645, A326647.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 17 2019