A356227 Smallest size of a maximal 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, 1, 1, 1, 1, 4, 2, 1, 3, 1, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Aug 13 2022

Keywords

• nonn

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 submultisets: {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.

Examples

			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) = 1.
		

Crossrefs

Positions of first appearances are A000079.

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

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

The greatest instead of smallest size is A356228.

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.

A356224 counts even gapless divisors, complement A356225.

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

Programs

Mathematica

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

Formula

a(n) = A333768(A356230(n)).

a(n) = A055396(A356231(n)).