A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

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

Offset

1,3

Comments

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.

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).

Links

Table of n, a(n) for n=1..81.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

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

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[If[n==1, 0, Max@@aug[Reverse[primeMS[n]]]], {n, 100}]

Crossrefs

Crossrefs found in the link are not repeated here.

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

Positions of first appearances are A008578.

Positions of 2's are A065119.

The non-augmented version is A286470, also A355526.

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

The minimal version is A355531.

Row maxima of A355534, which has Heinz number A325351.

A001222 counts prime indices, distinct A001221.

A112798 lists prime indices, sum A056239.

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

Sequence in context: A108230 A334202 A324729 * A364192 A253558 A061395

Adjacent sequences: A355529 A355530 A355531 * A355533 A355534 A355535

Keyword

nonn

Author

Gus Wiseman, Jul 14 2022

Status

approved