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A325405 Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

%I A325405 #7 Jun 07 2019 16:33:56
%S A325405 1,2,3,5,7,10,11,13,14,15,17,19,22,23,26,29,31,33,34,35,37,38,39,41,
%T A325405 43,46,47,51,53,55,57,58,59,61,62,67,69,71,73,74,77,79,82,83,85,86,87,
%U A325405 89,91,93,94,95,97,101,103,106,107,109,111,113,115,118,119,122
%N A325405 Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.
%C A325405 First differs from A325388 in lacking 130.
%C A325405 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325405 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 A325405 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 A325405 The enumeration of these partitions by sum is given by A325404.
%H A325405 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325405 The sequence of terms together with their prime indices begins:
%e A325405     1: {}
%e A325405     2: {1}
%e A325405     3: {2}
%e A325405     5: {3}
%e A325405     7: {4}
%e A325405    10: {1,3}
%e A325405    11: {5}
%e A325405    13: {6}
%e A325405    14: {1,4}
%e A325405    15: {2,3}
%e A325405    17: {7}
%e A325405    19: {8}
%e A325405    22: {1,5}
%e A325405    23: {9}
%e A325405    26: {1,6}
%e A325405    29: {10}
%e A325405    31: {11}
%e A325405    33: {2,5}
%e A325405    34: {1,7}
%e A325405    35: {3,4}
%t A325405 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325405 Select[Range[100],UnsameQ@@Join@@Table[Differences[primeMS[#],k],{k,0,PrimeOmega[#]}]&]
%Y A325405 A subsequence of A005117.
%Y A325405 Cf. A056239, A112798, A279945, A325325, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325404, A325406, A325467.
%K A325405 nonn
%O A325405 1,2
%A A325405 _Gus Wiseman_, May 02 2019