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A325368
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Heinz numbers of integer partitions with distinct differences between successive parts.
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23
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1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83
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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 enumeration of these partitions by sum is given by A325325.
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LINKS
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EXAMPLE
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Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
8: {1,1,1}
16: {1,1,1,1}
24: {1,1,1,2}
27: {2,2,2}
30: {1,2,3}
32: {1,1,1,1,1}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
72: {1,1,1,2,2}
80: {1,1,1,1,3}
81: {2,2,2,2}
88: {1,1,1,5}
90: {1,2,2,3}
96: {1,1,1,1,1,2}
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], UnsameQ@@Differences[primeptn[#]]&]
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CROSSREFS
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Cf. A056239, A112798, A130091, A240026, A325325, A325328, A325352, A325360, A325361, A325366, A325367, A325405, A325456, A325457.
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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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