%I
%S 1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,19,20,21,22,23,24,26,28,29,
%T 30,31,32,33,34,37,38,39,40,41,42,43,44,46,47,48,51,52,53,55,56,57,58,
%U 59,60,61,62,64,65,66,67,68,69,71,73,74,76,78,79,80,82,83
%N Heinz numbers of integer partitions whose augmented differences are weakly decreasing.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i  y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
%C The enumeration of these partitions by sum is given by A325350.
%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 6: {1,2}
%e 7: {4}
%e 8: {1,1,1}
%e 10: {1,3}
%e 11: {5}
%e 12: {1,1,2}
%e 13: {6}
%e 14: {1,4}
%e 15: {2,3}
%e 16: {1,1,1,1}
%e 17: {7}
%e 19: {8}
%e 20: {1,1,3}
%e 21: {2,4}
%e 22: {1,5}
%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t aug[y_]:=Table[If[i<Length[y],y[[i]]y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t Select[Range[100],GreaterEqual@@aug[primeptn[#]]&]
%Y Cf. A056239, A093641, A112798, A320466, A320509, A325350, A325351, A325361, A325364, A325366, A325394, A325395, A325396, A325397.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 02 2019
