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A325360
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Heinz numbers of integer partitions whose differences are weakly increasing.
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17
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}
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MATHEMATICA
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primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], OrderedQ[Differences[primeptn[#]]]&]
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CROSSREFS
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Cf. A007294, A056239, A112798, A240026, A325328, A325352, A325354, A325360, A325361, A325368, A325394, A325400.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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