

A325397


Heinz numbers of integer partitions whose kth differences are weakly decreasing for all k >= 0.


8



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89
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OFFSET

1,2


COMMENTS

First differs from A325361 in lacking 150.
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 A325353.


LINKS

Table of n, a(n) for n=1..67.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.


EXAMPLE

Most small numbers are in the sequence. However, the sequence of nonterms together with their prime indices begins:
12: {1,1,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
52: {1,1,6}
56: {1,1,1,4}
60: {1,1,2,3}
63: {2,2,4}
66: {1,2,5}
68: {1,1,7}
72: {1,1,1,2,2}
76: {1,1,8}
78: {1,2,6}
80: {1,1,1,1,3}
The first partition that has weakly decreasing differences (A320466, A325361) but is not represented in this sequence is (3,3,2,1), which has Heinz number 150 and whose first and second differences are (0,1,1) and (1,0) respectively.


MATHEMATICA

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], And@@Table[GreaterEqual@@Differences[primeptn[#], k], {k, 0, PrimeOmega[#]}]&]


CROSSREFS

Cf. A056239, A112798, A320466, A320509, A325353, A325361, A325364, A325389, A325398, A325399, A325400, A325405, A325467.
Sequence in context: A065200 A342526 A325361 * A289812 A206551 A338190
Adjacent sequences: A325394 A325395 A325396 * A325398 A325399 A325400


KEYWORD

nonn


AUTHOR

Gus Wiseman, May 02 2019


STATUS

approved



