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A325389 Heinz numbers of integer partitions whose augmented differences are weakly decreasing.

%I A325389 #4 May 03 2019 08:35:43
%S A325389 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 A325389 30,31,32,33,34,37,38,39,40,41,42,43,44,46,47,48,51,52,53,55,56,57,58,
%U A325389 59,60,61,62,64,65,66,67,68,69,71,73,74,76,78,79,80,82,83
%N A325389 Heinz numbers of integer partitions whose augmented differences are weakly decreasing.
%C A325389 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325389 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 A325389 The enumeration of these partitions by sum is given by A325350.
%H A325389 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325389 The sequence of terms together with their prime indices begins:
%e A325389    1: {}
%e A325389    2: {1}
%e A325389    3: {2}
%e A325389    4: {1,1}
%e A325389    5: {3}
%e A325389    6: {1,2}
%e A325389    7: {4}
%e A325389    8: {1,1,1}
%e A325389   10: {1,3}
%e A325389   11: {5}
%e A325389   12: {1,1,2}
%e A325389   13: {6}
%e A325389   14: {1,4}
%e A325389   15: {2,3}
%e A325389   16: {1,1,1,1}
%e A325389   17: {7}
%e A325389   19: {8}
%e A325389   20: {1,1,3}
%e A325389   21: {2,4}
%e A325389   22: {1,5}
%t A325389 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325389 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325389 Select[Range[100],GreaterEqual@@aug[primeptn[#]]&]
%Y A325389 Cf. A056239, A093641, A112798, A320466, A320509, A325350, A325351, A325361, A325364, A325366, A325394, A325395, A325396, A325397.
%K A325389 nonn
%O A325389 1,2
%A A325389 _Gus Wiseman_, May 02 2019