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%I #5 May 03 2019 08:37:38
%S 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,31,33,34,35,37,38,39,
%T 41,43,46,47,51,53,55,57,58,59,61,62,65,67,69,70,71,73,74,77,79,82,83,
%U 85,86,87,89,91,93,94,95,97,101,103,106,107,109,111,113,115
%N Heinz numbers of integer partitions whose k-th differences are strictly decreasing for all k >= 0.
%C First differs from A167171 in having 70. First differs from A325398 in lacking 42.
%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) are (-3,-2).
%C The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
%C The enumeration of these partitions by sum is given by A325393.
%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 6: {1,2}
%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 21: {2,4}
%e 22: {1,5}
%e 23: {9}
%e 26: {1,6}
%e 29: {10}
%e 31: {11}
%e 33: {2,5}
%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t Select[Range[100],And@@Table[Greater@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]
%Y A subsequence of A005117.
%Y Cf. A056239, A112798, A320466, A320470, A325358, A325391, A325393, A325396, A325397, A325398, A325400, A325405, A325457, A325467.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 02 2019
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