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A325360
Heinz numbers of integer partitions whose differences are weakly increasing.
17
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73
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 (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A240026.
EXAMPLE
Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
18: {1,2,2}
36: {1,1,2,2}
50: {1,3,3}
54: {1,2,2,2}
70: {1,3,4}
72: {1,1,1,2,2}
75: {2,3,3}
90: {1,2,2,3}
98: {1,4,4}
100: {1,1,3,3}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], OrderedQ[Differences[primeptn[#]]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 02 2019
STATUS
approved