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A325396 Heinz numbers of integer partitions whose augmented differences are strictly decreasing.

%I A325396 #4 May 03 2019 08:37:13
%S A325396 1,2,3,5,6,7,10,11,13,14,17,19,21,22,23,26,29,31,33,34,37,38,39,41,42,
%T A325396 43,46,47,51,53,57,58,59,61,62,65,66,67,69,71,73,74,78,79,82,83,85,86,
%U A325396 87,89,93,94,95,97,101,102,103,106,107,109,111,113,114,115
%N A325396 Heinz numbers of integer partitions whose augmented differences are strictly decreasing.
%C A325396 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325396 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 A325396 The enumeration of these partitions by sum is given by A325358.
%H A325396 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325396 The sequence of terms together with their prime indices begins:
%e A325396     1: {}
%e A325396     2: {1}
%e A325396     3: {2}
%e A325396     5: {3}
%e A325396     6: {1,2}
%e A325396     7: {4}
%e A325396    10: {1,3}
%e A325396    11: {5}
%e A325396    13: {6}
%e A325396    14: {1,4}
%e A325396    17: {7}
%e A325396    19: {8}
%e A325396    21: {2,4}
%e A325396    22: {1,5}
%e A325396    23: {9}
%e A325396    26: {1,6}
%e A325396    29: {10}
%e A325396    31: {11}
%e A325396    33: {2,5}
%e A325396    34: {1,7}
%t A325396 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325396 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325396 Select[Range[100],Greater@@aug[primeptn[#]]&]
%Y A325396 A subsequence of A005117.
%Y A325396 Cf. A056239, A093641, A112798, A320466, A325351, A325358, A325366, A325389, A325393, A325394, A325395, A325457.
%K A325396 nonn
%O A325396 1,2
%A A325396 _Gus Wiseman_, May 02 2019