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A326623 Heinz numbers of integer partitions whose geometric mean is an integer. 25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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}
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.
Sequence in context: A191840 A214981 A322030 * A210576 A191848 A360613
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 14 2019
STATUS
approved

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)