A326019 Heinz numbers of non-knapsack partitions such that every non-singleton submultiset has a different sum.

12, 30, 40, 63, 70, 112, 154, 165, 198, 220, 273, 286, 325, 351, 352, 364, 442, 561, 595, 646, 714, 741, 748, 765, 832, 850, 874, 918, 931, 952, 988, 1045, 1173, 1254, 1334, 1425, 1495, 1539, 1564, 1653, 1672, 1771, 1794, 1798, 1900, 2139, 2176, 2204, 2254

Offset

1,1

Comments

A subsequence of A299729.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

An integer partition is knapsack if every distinct submultiset has a different sum.

Links

Table of n, a(n) for n=1..49.

Example

The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   30: {1,2,3}
   40: {1,1,1,3}
   63: {2,2,4}
   70: {1,3,4}
  112: {1,1,1,1,4}
  154: {1,4,5}
  165: {2,3,5}
  198: {1,2,2,5}
  220: {1,1,3,5}
  273: {2,4,6}
  286: {1,5,6}
  325: {3,3,6}
  351: {2,2,2,6}
  352: {1,1,1,1,1,5}
  364: {1,1,4,6}
  442: {1,6,7}
  561: {2,5,7}
  595: {3,4,7}
  646: {1,7,8}

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
Select[Range[1000], !UnsameQ@@hwt/@Divisors[#]&&UnsameQ@@hwt/@Select[Divisors[#], !PrimeQ[#]&]&]

Crossrefs

Cf. A108917, A275972, A299702, A299729, A304793.

Cf. A325780, A325862, A325863, A326016, A326018.

Sequence in context: A024364 A093507 A325802 * A334804 A328411 A333840

Adjacent sequences: A326016 A326017 A326018 * A326020 A326021 A326022

Keyword

nonn

Author

Gus Wiseman, Jun 03 2019

Status

approved