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

Author

Gus Wiseman, Jun 03 2019

Keywords

• nonn

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.

Examples

			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}
		

Crossrefs

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

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

Programs

Mathematica

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