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A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

%I A355534 #9 Jul 14 2022 17:23:31
%S A355534 1,2,1,1,3,2,1,4,1,1,1,1,2,3,1,5,2,1,1,6,4,1,2,2,1,1,1,1,7,1,2,1,8,3,
%T A355534 1,1,3,2,5,1,9,2,1,1,1,1,3,6,1,1,1,2,4,1,1,10,2,2,1,11,1,1,1,1,1,4,2,
%U A355534 7,1,2,3,1,2,1,1,12,8,1,5,2,3,1,1,1
%N A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.
%C A355534 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A355534 The augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)_i = q_i - q_{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
%C A355534 One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.
%H A355534 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%e A355534 Triangle begins:
%e A355534    2: 1
%e A355534    3: 2
%e A355534    4: 1 1
%e A355534    5: 3
%e A355534    6: 2 1
%e A355534    7: 4
%e A355534    8: 1 1 1
%e A355534    9: 1 2
%e A355534   10: 3 1
%e A355534   11: 5
%e A355534   12: 2 1 1
%e A355534   13: 6
%e A355534   14: 4 1
%e A355534   15: 2 2
%e A355534   16: 1 1 1 1
%e A355534 For example, the reversed prime indices of 825 are (5,3,3,2), which have augmented differences (3,1,2,2).
%t A355534 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355534 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A355534 Table[aug[Reverse[primeMS[n]]],{n,30}]
%Y A355534 Crossrefs found in the link are not repeated here.
%Y A355534 Row-lengths are A001222.
%Y A355534 Row-sums are A252464
%Y A355534 Other similar triangles are A287352, A091602.
%Y A355534 Constant rows have indices A307824.
%Y A355534 The Heinz numbers of the rows are A325351.
%Y A355534 Strict rows have indices A325366.
%Y A355534 Row minima are A355531, non-augmented A355524, also A355525.
%Y A355534 Row maxima are A355535, non-augmented A286470, also A355526.
%Y A355534 The non-augmented version is A355536, also A355533.
%Y A355534 A112798 lists prime indices, sum A056239.
%Y A355534 Cf. A066312, A124010, A129654, A243055, A243056, A325352, A325394, A355523, A355528, A355531.
%K A355534 nonn,tabf
%O A355534 2,2
%A A355534 _Gus Wiseman_, Jul 12 2022