A326645 Heinz numbers of integer partitions whose mean and geometric mean are both integers.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 46, 47, 49, 53, 57, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 183, 191, 193

Offset

1,1

Comments

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

The enumeration of these partitions by sum is given by A326641.

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}
   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}
   41: {13}
   43: {14}
   46: {1,9}
   47: {15}
   49: {4,4}

Mathematica

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

Crossrefs

Heinz numbers of partitions with integer mean are A316413.

Heinz numbers of partitions with integer geometric mean are A326623.

Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.

Partitions with integer mean and geometric mean are A326641.

Subsets with integer mean and geometric mean are A326643.

Strict partitions with integer mean and geometric mean are A326029.

Cf. A051293, A056239, A067538, A067539, A078175, A112798, A316413, A326623, A326644, A326646, A326647.

Sequence in context: A268082 A247199 A087441 * A371445 A325371 A373071

Adjacent sequences: A326642 A326643 A326644 * A326646 A326647 A326648

Keyword

nonn

Author

Gus Wiseman, Jul 16 2019

Status

approved