cp's OEIS Frontend

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

%I A216650 #20 Nov 09 2022 19:08:59
%S A216650 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,
%T A216650 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,
%U A216650 3,3,2,1,1,2,2,1,1,2,2,1,3,6,1,5,2,2,2
%N A216650 Maximum length of each subsequence whose elements are the greatest prime divisors of the integers 2, 3, 4, ... in increasing order.
%C A216650 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, ...}.
%C A216650 See the examples for the structure of the subsequences of increasing prime divisors.
%C A216650 The growth of a(n) is very slow. See the following smallest values of m such that a(m) = n:
%C A216650 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.
%H A216650 Michel Lagneau, <a href="/A216650/b216650.txt">Table of n, a(n) for n = 1..10000</a>
%F A216650 a(n) = A070087(n)-A070087(n-1) for n >= 2. - _Pontus von Brömssen_, Nov 09 2022
%e A216650 Subset 1: {2, 3} obtained with the numbers 2, 3 => a(1) = 2;
%e A216650 Subset 2: {2, 5} obtained with the numbers 4, 5 => a(2) = 2;
%e A216650 Subset 3: {3, 7} obtained with the numbers 6, 7 => a(3) = 2;
%e A216650 Subset 4: {2, 3, 5, 11} obtained with the numbers 8, 9, 10, 11 => a(4) = 4.
%p A216650 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:
%Y A216650 Cf. A006530, A070087, A216651.
%K A216650 nonn
%O A216650 1,1
%A A216650 _Michel Lagneau_, Sep 12 2012