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Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).
6

%I #6 May 03 2019 08:35:34

%S 1,2,3,5,7,10,11,13,14,15,17,19,22,23,26,29,31,33,34,35,37,38,39,41,

%T 43,46,47,51,53,55,57,58,59,61,62,67,69,71,73,74,77,79,82,83,85,86,87,

%U 89,91,93,94,95,97,101,103,106,107,109,111,113,115,118,119,122

%N Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).

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

%C The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).

%C The enumeration of these partitions by sum is given by A320348.

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

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

%e 1: {}

%e 2: {1}

%e 3: {2}

%e 5: {3}

%e 7: {4}

%e 10: {1,3}

%e 11: {5}

%e 13: {6}

%e 14: {1,4}

%e 15: {2,3}

%e 17: {7}

%e 19: {8}

%e 22: {1,5}

%e 23: {9}

%e 26: {1,6}

%e 29: {10}

%e 31: {11}

%e 33: {2,5}

%e 34: {1,7}

%e 35: {3,4}

%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t Select[Range[100],SquareFreeQ[#]&&UnsameQ@@Differences[Append[primeptn[#],0]]&]

%Y A subsequence of A005117.

%Y Cf. A056239, A112798, A320348, A325324, A325327, A325362, A325364, A325366, A325367, A325368, A325390, A325405, A325460, A325461, A325467.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 02 2019