

A325398


Heinz numbers of reversed integer partitions whose kth 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 kth differences for k > 0 are the differences of the (k1)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



