A325461 - OEIS

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A325461

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

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 75, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197

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

The enumeration of these partitions by sum is given by A320510.

LINKS

Table of n, a(n) for n=1..61.

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}

7: {4}

9: {2,2}

11: {5}

13: {6}

15: {2,3}

17: {7}

19: {8}

23: {9}

25: {3,3}

29: {10}

31: {11}

35: {3,4}

37: {12}

41: {13}

43: {14}

MATHEMATICA

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

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

CROSSREFS

Cf. A056239, A112798, A320510, A325327, A325362, A325364, A325367, A325388, A325390, A325396, A325399, A325407, A325457, A325460.

Sequence in context: A032515 A024926 A051532 * A135785 A262249 A248421

Adjacent sequences: A325458 A325459 A325460 * A325462 A325463 A325464

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 03 2019

STATUS

approved