A216651 Lengths of decreasing blocks of A006530, the greatest prime factor of n, starting from the second term.

1, 2, 2, 2, 1, 1, 2, 4, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 3, 4, 2, 3, 1, 2, 2, 2, 2, 2, 1, 1, 2, 4, 2, 2, 2, 2, 3, 2, 1, 3, 1, 2, 1, 3, 2, 1, 1, 1, 3, 2, 2, 2, 3, 1, 2, 2, 2, 2, 1, 2, 3, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 3, 1, 2, 1, 2, 4, 2, 4, 2, 3, 1, 2, 1

Offset

1,2

Comments

Let gpf(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, ...}.

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

a(1) = 1, a(2) = 2, a(20) = 3, a(8) = 4, a(251) = 5, a(936) = 6, a(15553) = 7, a(6380) = 8, a(54838)=9, a(293548) = 10.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A070089(n+1)-A070089(n). - Pontus von Brömssen, Nov 09 2022

Example

A006530 with decreasing blocks marked: (2), (3, 2), (5, 3), (7, 2), (3), (5), (11, 3), (13, 7, 5, 2), .... Thus the terms of this sequence are 1, 2, 2, 2, 1, 1, 2, 4, ....

Maple

N:= 1000: # to use A006530(1..N)
L:= map(max @ numtheory:-factorset, [$1..N]):
DL:= L[2..-1]-L[1..-2]:
R:= select(t -> DL[t]>= 0, [$1..N-1]):
R[2..-1]-R[1..-2]; # Robert Israel, Mar 02 2018

Crossrefs

Cf. A006530, A216650.

First differences of A070089.

Sequence in context: A171412 A248213 A317682 * A071338 A078826 A051950

Adjacent sequences: A216648 A216649 A216650 * A216652 A216653 A216654

Keyword

nonn

Author

Michel Lagneau, Sep 12 2012

Status

approved