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A325457 Heinz numbers of integer partitions with strictly decreasing differences. 9
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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..65.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

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[#]]&]

CROSSREFS

Cf. A056239, A112798, A320470, A320510, A325328, A325352, A325360, A325361, A325368, A325399, A325456, A325461, A320470, A325396.

Sequence in context: A131511 A210490 A166155 * A063538 A167207 A037143

Adjacent sequences:  A325454 A325455 A325456 * A325458 A325459 A325460

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 03 2019

STATUS

approved

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Last modified April 5 22:24 EDT 2020. Contains 333260 sequences. (Running on oeis4.)