A325388 - OEIS

A325388

Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122

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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) (with the last part taken to be 0) are (-3,-2,-1).

The enumeration of these partitions by sum is given by A320348.

LINKS

Table of n, a(n) for n=1..64.

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}

7: {4}

10: {1,3}

11: {5}

13: {6}

14: {1,4}

15: {2,3}

17: {7}

19: {8}

22: {1,5}

23: {9}

26: {1,6}

29: {10}

31: {11}

33: {2,5}

34: {1,7}

35: {3,4}

MATHEMATICA

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

Select[Range[100], SquareFreeQ[#]&&UnsameQ@@Differences[Append[primeptn[#], 0]]&]

CROSSREFS

A subsequence of A005117.

Cf. A056239, A112798, A320348, A325324, A325327, A325362, A325364, A325366, A325367, A325368, A325390, A325405, A325460, A325461, A325467.

Sequence in context: A087246 A090421 A109608 * A325405 A118241 A356237

Adjacent sequences: A325385 A325386 A325387 * A325389 A325390 A325391

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved