A216650 -id:A216650 - cp's OEIS frontend

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

Michel Lagneau, Sep 12 2012

nonn

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.

			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, ....

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

Cf. A006530, A216650.

First differences of A070089.

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

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

cp's OEIS Frontend

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

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