A355534 - OEIS

%I #9 Jul 14 2022 17:23:31

%S 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 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 7,1,2,3,1,2,1,1,12,8,1,5,2,3,1,1,1

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

%C 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 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 One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.

%e Triangle begins:

%e    2: 1

%e    3: 2

%e    4: 1 1

%e    5: 3

%e    6: 2 1

%e    7: 4

%e    8: 1 1 1

%e    9: 1 2

%e   10: 3 1

%e   11: 5

%e   12: 2 1 1

%e   13: 6

%e   14: 4 1

%e   15: 2 2

%e   16: 1 1 1 1

%e For example, the reversed prime indices of 825 are (5,3,3,2), which have augmented differences (3,1,2,2).

%t primeMS[n_]:=If[n==1,{},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 Table[aug[Reverse[primeMS[n]]],{n,30}]

%Y Crossrefs found in the link are not repeated here.

%Y Row-lengths are A001222.

%Y Row-sums are A252464

%Y Other similar triangles are A287352, A091602.

%Y Constant rows have indices A307824.

%Y The Heinz numbers of the rows are A325351.

%Y Strict rows have indices A325366.

%Y Row minima are A355531, non-augmented A355524, also A355525.

%Y Row maxima are A355535, non-augmented A286470, also A355526.

%Y The non-augmented version is A355536, also A355533.

%Y A112798 lists prime indices, sum A056239.

%Y Cf. A066312, A124010, A129654, A243055, A243056, A325352, A325394, A355523, A355528, A355531.

%K nonn,tabf

%O 2,2

%A _Gus Wiseman_, Jul 12 2022