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Number of subsets of {1..n} whose mean and geometric mean are both integers.
12

%I #12 Aug 03 2019 22:19:36

%S 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 47,70,71,72,73,74,102,103,104,105,143,144,145,146,151,152,153,154,

%U 155,161,162,163,244,252,280,281,282,283,409,410,416,417,418,419

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

%H David Wasserman, <a href="/A326643/b326643.txt">Table of n, a(n) for n = 0..119</a>

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

%e The a(1) = 1 through a(12) = 16 subsets:

%e {1} {1} {1} {1} {1} {1} {1} {1} {1} {1} {1} {1}

%e {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2}

%e {3} {3} {3} {3} {3} {3} {3} {3} {3} {3}

%e {4} {4} {4} {4} {4} {4} {4} {4} {4}

%e {5} {5} {5} {5} {5} {5} {5} {5}

%e {6} {6} {6} {6} {6} {6} {6}

%e {7} {7} {7} {7} {7} {7}

%e {8} {8} {8} {8} {8}

%e {2,8} {9} {9} {9} {9}

%e {1,9} {10} {10} {10}

%e {2,8} {1,9} {11} {11}

%e {2,8} {1,9} {12}

%e {2,8} {1,9}

%e {2,8}

%e {3,6,12}

%e {3,4,9,12}

%t Table[Length[Select[Subsets[Range[n]],IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,0,10}]

%Y Partial sums of A326644.

%Y Subsets whose geometric mean is an integer are A326027.

%Y Subsets whose mean is an integer are A051293.

%Y Partitions with integer mean and geometric mean are A326641.

%Y Strict partitions with integer mean and geometric mean are A326029.

%Y Cf. A067538, A067539, A078175, A082553, A102627, A326623, A326625, A326645, A326646, A326647.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 16 2019

%E More terms from _David Wasserman_, Aug 03 2019