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

Offset: 1

Author

Gus Wiseman, Aug 16 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.

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.

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

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.

Programs

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}]

PARI

A287170(n) = { my(f=factor(n)); if(#f~==0, return (0), return(#f~ - sum(i=1, #f~-1, if (primepi(f[i, 1])+1 == primepi(f[i+1, 1]), 1, 0)))); };
A356229(n) = A287170(2*n); \\ Antti Karttunen, Jan 19 2025

Formula

a(n) = A287170(2n).

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 19 2025