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A326848
Heinz numbers of integer partitions of m >= 0 whose length times maximum is a multiple of m.
12
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 78, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 171, 173, 179, 181
OFFSET
1,2
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 A326849.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
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}
28: {1,1,4}
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], #==1||Divisible[Max[primeMS[#]]*Length[primeMS[#]], Total[primeMS[#]]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 26 2019
STATUS
approved