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Heinz numbers of integer partitions with strictly decreasing differences (with the last part taken to be 0).
7

%I #7 May 03 2019 21:26:15

%S 1,2,3,4,5,7,9,11,13,15,17,19,23,25,29,31,35,37,41,43,47,49,53,55,59,

%T 61,67,71,73,75,77,79,83,89,91,97,101,103,107,109,113,119,121,127,131,

%U 137,139,143,149,151,157,163,167,169,173,179,181,187,191,193,197

%N Heinz numbers of integer partitions with strictly decreasing 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 A320510.

%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 4: {1,1}

%e 5: {3}

%e 7: {4}

%e 9: {2,2}

%e 11: {5}

%e 13: {6}

%e 15: {2,3}

%e 17: {7}

%e 19: {8}

%e 23: {9}

%e 25: {3,3}

%e 29: {10}

%e 31: {11}

%e 35: {3,4}

%e 37: {12}

%e 41: {13}

%e 43: {14}

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

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

%Y Cf. A056239, A112798, A320510, A325327, A325362, A325364, A325367, A325388, A325390, A325396, A325399, A325407, A325457, A325460.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 03 2019