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A325366 Heinz numbers of integer partitions whose augmented differences are distinct.

%I A325366 #7 Apr 07 2021 12:22:44
%S A325366 1,2,3,5,6,7,9,10,11,13,14,17,19,21,22,23,25,26,29,31,33,34,35,37,38,
%T A325366 39,41,42,43,46,47,49,51,53,57,58,59,61,62,63,65,66,67,69,70,71,73,74,
%U A325366 77,78,79,82,83,85,86,87,89,91,93,94,95,97,99,101,102,103
%N A325366 Heinz numbers of integer partitions whose augmented differences are distinct.
%C A325366 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325366 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 A325366 The enumeration of these partitions by sum is given by A325349.
%H A325366 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325366 The sequence of terms together with their prime indices begins:
%e A325366     1: {}
%e A325366     2: {1}
%e A325366     3: {2}
%e A325366     5: {3}
%e A325366     6: {1,2}
%e A325366     7: {4}
%e A325366     9: {2,2}
%e A325366    10: {1,3}
%e A325366    11: {5}
%e A325366    13: {6}
%e A325366    14: {1,4}
%e A325366    17: {7}
%e A325366    19: {8}
%e A325366    21: {2,4}
%e A325366    22: {1,5}
%e A325366    23: {9}
%e A325366    25: {3,3}
%e A325366    26: {1,6}
%e A325366    29: {10}
%e A325366    31: {11}
%t A325366 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325366 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325366 Select[Range[100],UnsameQ@@aug[primeptn[#]]&]
%Y A325366 Positions of squarefree numbers in A325351.
%Y A325366 Cf. A056239, A093641, A112798, A130091, A325349, A325355, A325367, A325368, A325389, A325394, A325395, A325396, A325405.
%K A325366 nonn
%O A325366 1,2
%A A325366 _Gus Wiseman_, May 02 2019