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A325389 Heinz numbers of integer partitions whose augmented differences are weakly decreasing. 14
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 80, 82, 83 (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 augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The enumeration of these partitions by sum is given by A325350.

LINKS

Table of n, a(n) for n=1..67.

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}

   8: {1,1,1}

  10: {1,3}

  11: {5}

  12: {1,1,2}

  13: {6}

  14: {1,4}

  15: {2,3}

  16: {1,1,1,1}

  17: {7}

  19: {8}

  20: {1,1,3}

  21: {2,4}

  22: {1,5}

MATHEMATICA

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

aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];

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

CROSSREFS

Cf. A056239, A093641, A112798, A320466, A320509, A325350, A325351, A325361, A325364, A325366, A325394, A325395, A325396, A325397.

Sequence in context: A043094 A023803 A122132 * A020662 A306202 A328335

Adjacent sequences:  A325386 A325387 A325388 * A325390 A325391 A325392

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved

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Last modified December 11 02:16 EST 2019. Contains 329910 sequences. (Running on oeis4.)