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A326019
Heinz numbers of non-knapsack partitions such that every non-singleton submultiset has a different sum.
0
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.
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[#]&]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 03 2019
STATUS
approved