A355532 - OEIS

%I #8 Jul 14 2022 17:23:23

%S 0,1,2,1,3,2,4,1,2,3,5,2,6,4,2,1,7,2,8,3,3,5,9,2,3,6,2,4,10,2,11,1,4,

%T 7,3,2,12,8,5,3,13,3,14,5,2,9,15,2,4,3,6,6,16,2,3,4,7,10,17,2,18,11,3,

%U 1,4,4,19,7,8,3,20,2,21,12,2,8,4,5,22,3,2

%N Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

%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).

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

%e The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 3.

%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[If[n==1,0,Max@@aug[Reverse[primeMS[n]]]],{n,100}]

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

%Y Prepending 1 to the positions of 1's gives A000079.

%Y Positions of first appearances are A008578.

%Y Positions of 2's are A065119.

%Y The non-augmented version is A286470, also A355526.

%Y The non-augmented minimal version is A355524, also A355525.

%Y The minimal version is A355531.

%Y Row maxima of A355534, which has Heinz number A325351.

%Y A001222 counts prime indices, distinct A001221.

%Y A112798 lists prime indices, sum A056239.

%Y Cf. A066312, A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355533, A355536.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 14 2022