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A326027 Number of nonempty subsets of {1..n} whose geometric mean is an integer.

%I A326027 #17 Mar 07 2025 06:33:04
%S A326027 0,1,2,3,6,7,8,9,12,19,20,21,28,29,30,31,40,41,70,71,74,75,76,77,108,
%T A326027 123,124,211,214,215,216,217,332,333,334,335,592,593,594,595,612,613,
%U A326027 614,615,618,639,640,641,1160,1183,1324,1325,1328,1329,2176,2177,2196,2197,2198,2199,2414,2415,2416,2443,4000,4001,4002,4003,4006,4007,4008,4009,6626,6627,6628,9753,9756,9757,9758,9759,11136
%N A326027 Number of nonempty subsets of {1..n} whose geometric mean is an integer.
%H A326027 Max Alekseyev, <a href="/A326027/b326027.txt">Table of n, a(n) for n = 0..255</a>
%H A326027 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%F A326027 a(n) = A357413(n) + A357414(n). For a squarefree n, a(n) = a(n-1) + 1. - _Max Alekseyev_, Mar 01 2025
%e A326027 The a(1) = 1 through a(9) = 19 subsets:
%e A326027   {1}  {1}  {1}  {1}      {1}      {1}      {1}      {1}      {1}
%e A326027        {2}  {2}  {2}      {2}      {2}      {2}      {2}      {2}
%e A326027             {3}  {3}      {3}      {3}      {3}      {3}      {3}
%e A326027                  {4}      {4}      {4}      {4}      {4}      {4}
%e A326027                  {1,4}    {5}      {5}      {5}      {5}      {5}
%e A326027                  {1,2,4}  {1,4}    {6}      {6}      {6}      {6}
%e A326027                           {1,2,4}  {1,4}    {7}      {7}      {7}
%e A326027                                    {1,2,4}  {1,4}    {8}      {8}
%e A326027                                             {1,2,4}  {1,4}    {9}
%e A326027                                                      {2,8}    {1,4}
%e A326027                                                      {1,2,4}  {1,9}
%e A326027                                                      {2,4,8}  {2,8}
%e A326027                                                               {4,9}
%e A326027                                                               {1,2,4}
%e A326027                                                               {1,3,9}
%e A326027                                                               {2,4,8}
%e A326027                                                               {3,8,9}
%e A326027                                                               {4,6,9}
%e A326027                                                               {3,6,8,9}
%t A326027 Table[Length[Select[Subsets[Range[n]],IntegerQ[GeometricMean[#]]&]],{n,0,10}]
%Y A326027 First differences are A082553.
%Y A326027 Partitions whose geometric mean is an integer are A067539.
%Y A326027 Strict partitions whose geometric mean is an integer are A326625.
%Y A326027 Subsets whose average is an integer are A051293.
%Y A326027 Cf. A078174, A078175, A102627, A326567/A326568, A326622, A326623, A326624.
%K A326027 nonn
%O A326027 0,3
%A A326027 _Gus Wiseman_, Jul 14 2019
%E A326027 Terms a(57) onward from _Max Alekseyev_, Mar 01 2025