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A325467 Heinz numbers of integer partitions y such that the k-th differences of y are distinct (independently) for all k >= 0.

%I A325467 #14 Jun 17 2019 17:40:13
%S A325467 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,31,33,34,35,37,38,39,
%T A325467 41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,77,78,
%U A325467 79,82,83,85,86,87,89,91,93,94,95,97,101,102,103,106,107
%N A325467 Heinz numbers of integer partitions y such that the k-th differences of y are distinct (independently) for all k >= 0.
%C A325467 First differs from A301899 in having 70 and lacking 105.
%C A325467 First differs from A325398 in having 70.
%C A325467 First differs from A319315 in having 966.
%C A325467 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325467 The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
%C A325467 The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
%C A325467 The enumeration of these partitions by sum is given by A325468.
%H A325467 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325467 The sequence of terms together with their prime indices begins:
%e A325467     1: {}
%e A325467     2: {1}
%e A325467     3: {2}
%e A325467     5: {3}
%e A325467     6: {1,2}
%e A325467     7: {4}
%e A325467    10: {1,3}
%e A325467    11: {5}
%e A325467    13: {6}
%e A325467    14: {1,4}
%e A325467    15: {2,3}
%e A325467    17: {7}
%e A325467    19: {8}
%e A325467    21: {2,4}
%e A325467    22: {1,5}
%e A325467    23: {9}
%e A325467    26: {1,6}
%e A325467    29: {10}
%e A325467    31: {11}
%e A325467    33: {2,5}
%e A325467 For example, the k-th differences for k = 0...3 of the partition (9,4,2,1) with Heinz number 966 are
%e A325467    9  4  2  1
%e A325467   -5 -2 -1
%e A325467    3  1
%e A325467   -2
%e A325467 and since the entries of each row are distinct, 966 belongs to the sequence.
%t A325467 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325467 Select[Range[100],And@@Table[UnsameQ@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]
%Y A325467 A subsequence of A005117.
%Y A325467 Cf. A056239, A112798, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325405, A325468.
%K A325467 nonn
%O A325467 1,2
%A A325467 _Gus Wiseman_, May 03 2019