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
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 16 2019
STATUS
approved