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A325395 Heinz numbers of integer partitions whose augmented differences are strictly increasing.

%I A325395 #4 May 03 2019 08:37:05
%S A325395 1,2,3,5,7,9,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,59,61,67,71,
%T A325395 73,77,79,83,89,91,97,101,103,107,109,113,121,127,131,137,139,143,149,
%U A325395 151,157,163,167,169,173,179,181,187,191,193,197,199,209,211,221
%N A325395 Heinz numbers of integer partitions whose augmented differences are strictly increasing.
%C A325395 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325395 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 A325395 The enumeration of these partitions by sum is given by A325357.
%H A325395 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325395 The sequence of terms together with their prime indices begins:
%e A325395     1: {}
%e A325395     2: {1}
%e A325395     3: {2}
%e A325395     5: {3}
%e A325395     7: {4}
%e A325395     9: {2,2}
%e A325395    11: {5}
%e A325395    13: {6}
%e A325395    17: {7}
%e A325395    19: {8}
%e A325395    23: {9}
%e A325395    25: {3,3}
%e A325395    29: {10}
%e A325395    31: {11}
%e A325395    35: {3,4}
%e A325395    37: {12}
%e A325395    41: {13}
%e A325395    43: {14}
%e A325395    47: {15}
%e A325395    49: {4,4}
%t A325395 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325395 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325395 Select[Range[100],Less@@aug[primeptn[#]]&]
%Y A325395 Cf. A056239, A093641, A112798, A240027, A325351, A325357, A325366, A325389, A325394, A325396, A325398, A325456, A325460.
%K A325395 nonn
%O A325395 1,2
%A A325395 _Gus Wiseman_, May 02 2019