A325399 - OEIS

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A325399

Heinz numbers of integer partitions whose k-th differences are strictly decreasing for all k >= 0.

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, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

First differs from A167171 in having 70. First differs from A325398 in lacking 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 A325393.

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

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

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

CROSSREFS

A subsequence of A005117.

Cf. A056239, A112798, A320466, A320470, A325358, A325391, A325393, A325396, A325397, A325398, A325400, A325405, A325457, A325467.

Sequence in context: A325779 A301899 A325398 * A167171 A087008 A326537

Adjacent sequences: A325396 A325397 A325398 * A325400 A325401 A325402

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved