A326623 Heinz numbers of integer partitions whose geometric mean is an integer.

2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 42, 43, 46, 47, 49, 53, 57, 59, 61, 64, 67, 71, 73, 76, 79, 81, 83, 89, 97, 101, 103, 106, 107, 109, 113, 121, 125, 126, 127, 128, 131, 137, 139, 149, 151, 157, 161, 163, 167, 169

Offset

1,1

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Table of n, a(n) for n=1..61.

Wikipedia, Geometric mean

Example

The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], IntegerQ[GeometricMean[primeMS[#]]]&]

Crossrefs

The enumeration of these partitions by sum is given by A067539.

Heinz numbers of partitions with integer average are A316413.

The case without prime powers is A326624.

Subsets whose geometric mean is an integer are A326027.

Factorizations with integer geometric mean are A326028.

Cf. A001055, A078175, A102627, A326567/A326568, A326622, A326625.

Sequence in context: A376371 A214981 A322030 * A210576 A191848 A360613

Adjacent sequences: A326620 A326621 A326622 * A326624 A326625 A326626

Keyword

nonn

Author

Gus Wiseman, Jul 14 2019

Status

approved