A216650 Maximum length of each subsequence whose elements are the greatest prime divisors of the integers 2, 3, 4, ... in increasing order.

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

Offset: 1

Author

Michel Lagneau, Sep 12 2012

Keywords

• nonn

Comments

Let gpf(m) = A006530(m) be the greatest prime factor of m and the subset E(n) = {m, m+1, ..., m+L-1} such that gpf(m) < gpf(m+1) < ... < gpf(m+L-1) where L is the maximum length of E(n) and n the index such that {E(1) union E(2) union ... } = {2, 3, 4, ...}.

See the examples for the structure of the subsequences of increasing prime divisors.

The growth of a(n) is very slow. See the following smallest values of m such that a(m) = n:

a(6) = 1, a(1) = 2, a(11) = 3, a(4) = 4, a(44) = 5, a(82) = 6, a(4672) = 7, a(23001) = 8, a(360896) = 9.

Examples

			Subset 1: {2, 3} obtained with the numbers 2, 3 => a(1) = 2;
Subset 2: {2, 5} obtained with the numbers 4, 5 => a(2) = 2;
Subset 3: {3, 7} obtained with the numbers 6, 7 => a(3) = 2;
Subset 4: {2, 3, 5, 11} obtained with the numbers 8, 9, 10, 11 => a(4) = 4.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006530, A070087, A216651.

Programs

Maple

with(numtheory):p0:=2:it:=1:for n from 3 to 200 do: x:=factorset(n):n1:=nops(x):p:=x[n1]:if p>p0 then it:=it+1:p0:=p:else printf(`%d, `,it):it:=1:p0:=p:fi:od:

Formula

a(n) = A070087(n)-A070087(n-1) for n >= 2. - Pontus von Brömssen, Nov 09 2022