A356228 Greatest size of a gapless submultiset of the prime indices of n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 3, 1, 2, 1, 2, 3, 1, 1, 5, 2, 2, 1, 2, 1, 4, 1, 3, 1, 1, 1, 4, 1, 1, 2, 6, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 3, 2, 2, 2, 1, 4, 4, 1, 1, 3, 1, 1, 1

Offset

1,4

Comments

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless intervals: {2,3}, {5,5,6}, {9}.

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

Table of n, a(n) for n=1..87.

Formula

a(n) = A333766(A356230(n)).

a(n) = A061395(A356231(n)).

Example

The prime indices of 700 are {1,1,3,3,4}, with maximal gapless submultisets {1,1}, {3,3,4}, so a(700) = 3.
The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[n==1, 0, Max@@Length/@Split[primeMS[n], #1>=#2-1&]], {n, 100}]

Crossrefs

Positions of first appearances are A000079.

The maximal gapless submultisets are counted by A287170, firsts A066205.

These are the row-maxima of A356226, firsts A356232.

The smallest instead of greatest size is A356227.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

A001223 lists the prime gaps, reduced A028334.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

A073491 lists numbers with gapless prime indices, cf. A073492-A073495.

A356069 counts gapless divisors.

A356224 counts even gapless divisors, complement A356225.

Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A328162, A328457, A356229.

Sequence in context: A356226 A322872 A356227 * A284257 A318883 A356233

Adjacent sequences: A356225 A356226 A356227 * A356229 A356230 A356231

Keyword

nonn

Author

Gus Wiseman, Aug 13 2022

Status

approved