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

1,1

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.

Links

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

Formula

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

Example

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.

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:

Crossrefs

Cf. A006530, A070087, A216651.

Sequence in context: A335694 A084862 A327987 * A027387 A226534 A138260

Adjacent sequences: A216647 A216648 A216649 * A216651 A216652 A216653

Keyword

nonn

Author

Michel Lagneau, Sep 12 2012

Status

approved