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%I #7 Apr 07 2021 12:22:44
%S 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 39,41,42,43,46,47,49,51,53,57,58,59,61,62,63,65,66,67,69,70,71,73,74,
%U 77,78,79,82,83,85,86,87,89,91,93,94,95,97,99,101,102,103
%N Heinz numbers of integer partitions whose augmented differences are distinct.
%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 A325349.
%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 5: {3}
%e 6: {1,2}
%e 7: {4}
%e 9: {2,2}
%e 10: {1,3}
%e 11: {5}
%e 13: {6}
%e 14: {1,4}
%e 17: {7}
%e 19: {8}
%e 21: {2,4}
%e 22: {1,5}
%e 23: {9}
%e 25: {3,3}
%e 26: {1,6}
%e 29: {10}
%e 31: {11}
%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],UnsameQ@@aug[primeptn[#]]&]
%Y Positions of squarefree numbers in A325351.
%Y Cf. A056239, A093641, A112798, A130091, A325349, A325355, A325367, A325368, A325389, A325394, A325395, A325396, A325405.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 02 2019
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