

A325361


Heinz numbers of integer partitions whose differences are weakly decreasing.


12



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

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, for example the differences of (x, y, z) are (y  x, z  y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) are (3,2).
The enumeration of these partitions by sum is given by A320466.


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}


MATHEMATICA

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


CROSSREFS

Cf. A056239, A112798, A320466, A320509, A325328, A325352, A325456, A325457, A325360, A325361, A325364, A320466, A325368, A325389.
Sequence in context: A316529 A329138 A065200 * A325397 A289812 A206551
Adjacent sequences: A325358 A325359 A325360 * A325362 A325363 A325364


KEYWORD

nonn


AUTHOR

Gus Wiseman, May 02 2019


STATUS

approved



