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A326643 Number of subsets of {1..n} whose mean and geometric mean are both integers.

%I A326643 #12 Aug 03 2019 22:19:36
%S A326643 0,1,2,3,4,5,6,7,9,11,12,13,16,17,18,19,22,23,30,31,32,33,34,35,41,46,
%T A326643 47,70,71,72,73,74,102,103,104,105,143,144,145,146,151,152,153,154,
%U A326643 155,161,162,163,244,252,280,281,282,283,409,410,416,417,418,419
%N A326643 Number of subsets of {1..n} whose mean and geometric mean are both integers.
%H A326643 David Wasserman, <a href="/A326643/b326643.txt">Table of n, a(n) for n = 0..119</a>
%H A326643 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A326643 The a(1) = 1 through a(12) = 16 subsets:
%e A326643   {1}  {1}  {1}  {1}  {1}  {1}  {1}  {1}    {1}    {1}    {1}    {1}
%e A326643        {2}  {2}  {2}  {2}  {2}  {2}  {2}    {2}    {2}    {2}    {2}
%e A326643             {3}  {3}  {3}  {3}  {3}  {3}    {3}    {3}    {3}    {3}
%e A326643                  {4}  {4}  {4}  {4}  {4}    {4}    {4}    {4}    {4}
%e A326643                       {5}  {5}  {5}  {5}    {5}    {5}    {5}    {5}
%e A326643                            {6}  {6}  {6}    {6}    {6}    {6}    {6}
%e A326643                                 {7}  {7}    {7}    {7}    {7}    {7}
%e A326643                                      {8}    {8}    {8}    {8}    {8}
%e A326643                                      {2,8}  {9}    {9}    {9}    {9}
%e A326643                                             {1,9}  {10}   {10}   {10}
%e A326643                                             {2,8}  {1,9}  {11}   {11}
%e A326643                                                    {2,8}  {1,9}  {12}
%e A326643                                                           {2,8}  {1,9}
%e A326643                                                                  {2,8}
%e A326643                                                                  {3,6,12}
%e A326643                                                                  {3,4,9,12}
%t A326643 Table[Length[Select[Subsets[Range[n]],IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,0,10}]
%Y A326643 Partial sums of A326644.
%Y A326643 Subsets whose geometric mean is an integer are A326027.
%Y A326643 Subsets whose mean is an integer are A051293.
%Y A326643 Partitions with integer mean and geometric mean are A326641.
%Y A326643 Strict partitions with integer mean and geometric mean are A326029.
%Y A326643 Cf. A067538, A067539, A078175, A082553, A102627, A326623, A326625, A326645, A326646, A326647.
%K A326643 nonn
%O A326643 0,3
%A A326643 _Gus Wiseman_, Jul 16 2019
%E A326643 More terms from _David Wasserman_, Aug 03 2019