

A325405


Heinz numbers of integer partitions y such that the kth differences of y are distinct for all k >= 0 and are disjoint from the ith differences for i != k.


14



1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122
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OFFSET

1,2


COMMENTS

First differs from A325388 in lacking 130.
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 A325404.


LINKS

Table of n, a(n) for n=1..64.
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}
7: {4}
10: {1,3}
11: {5}
13: {6}
14: {1,4}
15: {2,3}
17: {7}
19: {8}
22: {1,5}
23: {9}
26: {1,6}
29: {10}
31: {11}
33: {2,5}
34: {1,7}
35: {3,4}


MATHEMATICA

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


CROSSREFS

A subsequence of A005117.
Cf. A056239, A112798, A279945, A325325, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325404, A325406, A325467.
Sequence in context: A090421 A109608 A325388 * A118241 A325160 A258613
Adjacent sequences: A325402 A325403 A325404 * A325406 A325407 A325408


KEYWORD

nonn


AUTHOR

Gus Wiseman, May 02 2019


STATUS

approved



