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A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0. 10
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

First differs from A301899 in lacking 105. First differs from A325399 in having 42.

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 zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.

The enumeration of these partitions by sum is given by A325391.

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}

    5: {3}

    6: {1,2}

    7: {4}

   10: {1,3}

   11: {5}

   13: {6}

   14: {1,4}

   15: {2,3}

   17: {7}

   19: {8}

   21: {2,4}

   22: {1,5}

   23: {9}

   26: {1,6}

   29: {10}

   31: {11}

   33: {2,5}

MATHEMATICA

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

Select[Range[100], And@@Table[Less@@Differences[primeMS[#], k], {k, 0, PrimeOmega[#]}]&]

CROSSREFS

A subsequence of A005117.

Cf. A056239, A112798, A325357, A325391, A325395, A325397, A325399, A325400, A325405, A325406, A325456, A325467.

Sequence in context: A325467 A325779 A301899 * A325399 A167171 A087008

Adjacent sequences:  A325395 A325396 A325397 * A325399 A325400 A325401

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved

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Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)