A356229 Number of maximal gapless submultisets of the prime indices of 2n.

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

Offset

1,5

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.

This is a bisection of A287170, but is important in its own right because the even numbers are exactly those whose prime indices begin with 1.

Links

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

Formula

a(n) = A287170(2n).

Example

The prime indices of 2*9282 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(9282) = 3.

Mathematica

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

Crossrefs

This is the even (bisected) case of A287170, firsts A066205.

Alternate row-lengths of A356226, minima A356227(2n), maxima A356228(2n).

A001221 counts distinct prime factors, sum A001414.

A001222 counts prime indices, listed by A112798, sum A056239.

A003963 multiplies together the prime indices of n.

A073093 counts the prime indices of 2n.

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

Cf. A000005, A060680, A060681, A132747, A132881, A286470, A289509, A356230, A356231, A356232.

Sequence in context: A106802 A269254 A049236 * A244259 A094840 A035218

Adjacent sequences: A356226 A356227 A356228 * A356230 A356231 A356232

Keyword

nonn

Author

Gus Wiseman, Aug 16 2022

Status

approved