

A325395


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


13



1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211, 221
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 A325357.


LINKS

Table of n, a(n) for n=1..60.
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}
9: {2,2}
11: {5}
13: {6}
17: {7}
19: {8}
23: {9}
25: {3,3}
29: {10}
31: {11}
35: {3,4}
37: {12}
41: {13}
43: {14}
47: {15}
49: {4,4}


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], Less@@aug[primeptn[#]]&]


CROSSREFS

Cf. A056239, A093641, A112798, A240027, A325351, A325357, A325366, A325389, A325394, A325396, A325398, A325456, A325460.
Sequence in context: A319151 A180934 A090332 * A070566 A325623 A136327
Adjacent sequences: A325392 A325393 A325394 * A325396 A325397 A325398


KEYWORD

nonn


AUTHOR

Gus Wiseman, May 02 2019


STATUS

approved



