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A325456 Heinz numbers of integer partitions with strictly increasing differences.

%I A325456 #5 May 03 2019 21:25:51
%S A325456 1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,19,20,21,22,23,25,26,28,29,31,
%T A325456 33,34,35,37,38,39,41,42,43,44,45,46,47,49,51,52,53,55,57,58,59,61,62,
%U A325456 63,65,66,67,68,69,71,73,74,76,77,78,79,82,83,84,85,86,87
%N A325456 Heinz numbers of integer partitions with strictly increasing differences.
%C A325456 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325456 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 A325456 The enumeration of these partitions by sum is given by A240027.
%H A325456 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325456 The sequence of terms together with their prime indices begins:
%e A325456    1: {}
%e A325456    2: {1}
%e A325456    3: {2}
%e A325456    4: {1,1}
%e A325456    5: {3}
%e A325456    6: {1,2}
%e A325456    7: {4}
%e A325456    9: {2,2}
%e A325456   10: {1,3}
%e A325456   11: {5}
%e A325456   12: {1,1,2}
%e A325456   13: {6}
%e A325456   14: {1,4}
%e A325456   15: {2,3}
%e A325456   17: {7}
%e A325456   19: {8}
%e A325456   20: {1,1,3}
%e A325456   21: {2,4}
%e A325456   22: {1,5}
%e A325456   23: {9}
%t A325456 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325456 Select[Range[100],Less@@Differences[primeptn[#]]&]
%Y A325456 Cf. A056239, A112798, A179269, A240026, A240027, A325328, A325352, A325360, A325361, A325368, A325395, A325398, A325457, A325460.
%K A325456 nonn
%O A325456 1,2
%A A325456 _Gus Wiseman_, May 03 2019