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A325457
Heinz numbers of integer partitions with strictly decreasing differences.
10
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98
OFFSET
1,2
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A320470.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
6: {1,2}
7: {4}
9: {2,2}
10: {1,3}
11: {5}
12: {1,1,2}
13: {6}
14: {1,4}
15: {2,3}
17: {7}
19: {8}
20: {1,1,3}
21: {2,4}
22: {1,5}
23: {9}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], Greater@@Differences[primeptn[#]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 03 2019
STATUS
approved