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A325986 Heinz numbers of complete strict integer partitions. 6

%I #7 May 31 2019 05:34:09

%S 1,2,6,30,42,210,330,390,462,510,546,714,798,2310,2730,3570,3990,4290,

%T 4830,5610,6006,6090,6270,6510,6630,7410,7590,7854,8778,8970,9282,

%U 9570,9690,10230,10374,10626,11310,11730,12090,12210,12558,13398,13566,14322,14430

%N Heinz numbers of complete strict integer partitions.

%C Strict partitions are counted by A000009, while complete partitions are counted by A126796.

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

%C An integer partition of n is complete (A126796, A325781) if every number from 0 to n is the sum of some submultiset of the parts.

%C The enumeration of these partitions by sum is given by A188431.

%F Intersection of A005117 (strict partitions) and A325781 (complete partitions).

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

%e 1: {}

%e 2: {1}

%e 6: {1,2}

%e 30: {1,2,3}

%e 42: {1,2,4}

%e 210: {1,2,3,4}

%e 330: {1,2,3,5}

%e 390: {1,2,3,6}

%e 462: {1,2,4,5}

%e 510: {1,2,3,7}

%e 546: {1,2,4,6}

%e 714: {1,2,4,7}

%e 798: {1,2,4,8}

%e 2310: {1,2,3,4,5}

%e 2730: {1,2,3,4,6}

%e 3570: {1,2,3,4,7}

%e 3990: {1,2,3,4,8}

%e 4290: {1,2,3,5,6}

%e 4830: {1,2,3,4,9}

%e 5610: {1,2,3,5,7}

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

%t Select[Range[1000],SquareFreeQ[#]&&Union[hwt/@Divisors[#]]==Range[0,hwt[#]]&]

%Y Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A304793.

%Y Cf. A325780, A325781, A325782, A325788, A325790, A325988.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 30 2019

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)