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Heinz numbers of non-knapsack partitions such that every non-singleton submultiset has a different sum.
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%I #5 Jun 04 2019 08:36:27

%S 12,30,40,63,70,112,154,165,198,220,273,286,325,351,352,364,442,561,

%T 595,646,714,741,748,765,832,850,874,918,931,952,988,1045,1173,1254,

%U 1334,1425,1495,1539,1564,1653,1672,1771,1794,1798,1900,2139,2176,2204,2254

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

%C A subsequence of A299729.

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

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

%e The sequence of terms together with their prime indices begins:

%e 12: {1,1,2}

%e 30: {1,2,3}

%e 40: {1,1,1,3}

%e 63: {2,2,4}

%e 70: {1,3,4}

%e 112: {1,1,1,1,4}

%e 154: {1,4,5}

%e 165: {2,3,5}

%e 198: {1,2,2,5}

%e 220: {1,1,3,5}

%e 273: {2,4,6}

%e 286: {1,5,6}

%e 325: {3,3,6}

%e 351: {2,2,2,6}

%e 352: {1,1,1,1,1,5}

%e 364: {1,1,4,6}

%e 442: {1,6,7}

%e 561: {2,5,7}

%e 595: {3,4,7}

%e 646: {1,7,8}

%t hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];

%t Select[Range[1000],!UnsameQ@@hwt/@Divisors[#]&&UnsameQ@@hwt/@Select[Divisors[#],!PrimeQ[#]&]&]

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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 03 2019