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editing

proposed

allocated for Gus WisemanHeinz numbers of integer partitions whose differences are weakly decreasing.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89

1,2

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, 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 A320466.

Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:

12: {1,1,2}

20: {1,1,3}

24: {1,1,1,2}

28: {1,1,4}

36: {1,1,2,2}

40: {1,1,1,3}

42: {1,2,4}

44: {1,1,5}

45: {2,2,3}

48: {1,1,1,1,2}

52: {1,1,6}

56: {1,1,1,4}

60: {1,1,2,3}

63: {2,2,4}

66: {1,2,5}

68: {1,1,7}

72: {1,1,1,2,2}

76: {1,1,8}

78: {1,2,6}

80: {1,1,1,1,3}

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];

Select[Range[100], GreaterEqual@@Differences[primeptn[#]]&]

Cf. A056239, A112798, A320466, A320509, A325328, A325352, A325456, A325457, A325360, A325361, A325364, A320466, A325368, A325389.

allocated

nonn

Gus Wiseman, May 02 2019

approved

editing

allocated for Gus Wiseman

allocated

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