A355526 Maximal difference between adjacent prime indices of n, or k if n is the k-th prime.

1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 1, 6, 3, 1, 0, 7, 1, 8, 2, 2, 4, 9, 1, 0, 5, 0, 3, 10, 1, 11, 0, 3, 6, 1, 1, 12, 7, 4, 2, 13, 2, 14, 4, 1, 8, 15, 1, 0, 2, 5, 5, 16, 1, 2, 3, 6, 9, 17, 1, 18, 10, 2, 0, 3, 3, 19, 6, 7, 2, 20, 1, 21, 11, 1, 7, 1, 4, 22, 2, 0, 12

Offset: 2

Author

Gus Wiseman, Jul 10 2022

Keywords

• nonn

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.

Examples

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 4.
		

Links

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

Crossrefs

Crossrefs found in the link are not repeated here.

Positions of first appearances are 4 followed by A000040.

Positions of 0's are A025475, minimal version A013929.

Positions of 1's are 2 followed by A066312, minimal version A355527.

Triangle A238710 counts m such that A056239(m) = n and a(m) = k.

Prepending 0 to the prime indices gives A286469, minimal version A355528.

See also A286470, minimal version A355524.

The minimal version is A355525, triangle A238709.

The augmented version is A355532.

A001522 counts partitions with a fixed point (unproved), ranked by A352827.

A287352, A355533, A355534, A355536 list the differences of prime indices.

Cf. A000005, A047966, A091602, A238353, A279945, A325160, A325161, A352822, A351294, A355530.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],PrimePi[n],Max@@Differences[primeMS[n]]],{n,2,100}]