A355523 Number of distinct differences between adjacent prime indices of n.

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 1, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 2, 0, 2, 0, 1, 2, 2, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 2, 1, 2, 1, 1, 1, 2, 0, 2, 2, 2, 0, 2, 0, 2, 1

Offset

1,12

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

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

Index entries for sequences related to prime indices in the factorization of n.

Example

For example, the prime indices of 22770 are {1,2,2,3,5,9}, with differences (1,0,1,2,4), so a(22770) = 4.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Union[Differences[primeMS[n]]]], {n, 1000}]

Prog

(PARI) A355523(n) = if(1==n, 0, my(pis = apply(primepi, factor(n)[, 1]), difs = vector(#pis-1, i, pis[i+1]-pis[i])); (#Set(difs)+!issquarefree(n))); \\ Antti Karttunen, Jan 20 2025

Crossrefs

Crossrefs found in the link are not repeated here.

Counting m such that A056239(m) = n and a(m) = k gives A279945.

With multiplicity we have A252736(n) = A001222(n) - 1.

The maximal difference is A286470, minimal A355524.

A008578 gives the positions of 0's.

A287352 lists differences between 0-prepended prime indices.

A355534 lists augmented differences between prime indices.

A355536 lists differences between prime indices.

Cf. A066312, A238353, A238710, A286469, A320348, A325388, A325406, A351294, A352827, A355525-A355533.

Sequence in context: A063962 A084114 A294881 * A110475 A366739 A086971

Adjacent sequences: A355520 A355521 A355522 * A355524 A355525 A355526

Keyword

nonn

Author

Gus Wiseman, Jul 10 2022

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

Status

approved