A325394 - OEIS

A325394

Heinz numbers of integer partitions whose augmented differences are weakly increasing.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 75, 77, 79, 81, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167

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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 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 A325356.

LINKS

Table of n, a(n) for n=1..62.

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}

8: {1,1,1}

9: {2,2}

11: {5}

13: {6}

15: {2,3}

16: {1,1,1,1}

17: {7}

19: {8}

23: {9}

25: {3,3}

27: {2,2,2}

29: {10}

31: {11}

32: {1,1,1,1,1}

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], OrderedQ[aug[primeptn[#]]]&]

CROSSREFS

Cf. A056239, A093641, A112798, A240026, A325351, A325356, A325360, A325362, A325366, A325389, A325395, A325396, A325400.

Sequence in context: A056867 A320324 A321698 * A358378 A062491 A087092

Adjacent sequences: A325391 A325392 A325393 * A325395 A325396 A325397

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved