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%I #5 May 03 2019 08:36:57
%S 1,2,3,4,5,7,8,9,11,13,15,16,17,19,23,25,27,29,31,32,35,37,41,43,47,
%T 49,53,55,59,61,64,67,71,73,75,77,79,81,83,89,91,97,101,103,105,107,
%U 109,113,119,121,125,127,128,131,137,139,143,149,151,157,163,167
%N Heinz numbers of integer partitions whose augmented differences are weakly increasing.
%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 A325356.
%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 7: {4}
%e 8: {1,1,1}
%e 9: {2,2}
%e 11: {5}
%e 13: {6}
%e 15: {2,3}
%e 16: {1,1,1,1}
%e 17: {7}
%e 19: {8}
%e 23: {9}
%e 25: {3,3}
%e 27: {2,2,2}
%e 29: {10}
%e 31: {11}
%e 32: {1,1,1,1,1}
%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],OrderedQ[aug[primeptn[#]]]&]
%Y Cf. A056239, A093641, A112798, A240026, A325351, A325356, A325360, A325362, A325366, A325389, A325395, A325396, A325400.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 02 2019
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