A331032 - OEIS

%I #27 May 04 2020 13:39:21

%S 0,0,1,0,2,0,4,2,0,1,6,0,10,3,0,4,12,3,16,0,2,5,18,2,0,9,1,1,22,0,28,

%T 12,4,11,0,9,30,15,8,5,36,0,40,3,4,17,42,11,0,3,10,7,46,15,2,0,14,21,

%U 52,7,58,27,0,2,6,1,60,9,16,0,66,11,70,29,10,13

%N Number of iterations of n -> n + gpf(n) needed for the trajectory of n to join the trajectory of A076271, where gpf(n) is the greatest prime factor of n.

%C Record values occur at prime values of n, and equal one less than the next lowest prime number (see Formula). Because of this, a(n) is always less than n, so for any positive integer starting value n, iterations of n -> n + gpf(n) will eventually join A076271.

%H Rémy Sigrist, <a href="/A331032/b331032.txt">Table of n, a(n) for n = 1..10000</a>

%F a(k*p) = prevprime(p) - k for all k <= prevprime(p).

%F a(p) = prevprime(p) - 1 for p > 2.

%e a(8)=2 because the trajectory for 1 (sequence A076271) starts 1->2->4->6->9->12->15->20... and the trajectory for 8 starts 8->10->15->20... so the sequence beginning with 8 joins A076271 after 2 steps.

%o (PARI) gpf(n) = if (n==1, 1, my (f=factor(n)); f[#f~, 1])

%o a(n) = { my (o=1); for (k=0, oo, while (o<n, o=o+gpf(o)); if (o==n, return (k), n=n+gpf(n))) } \\ _Rémy Sigrist_, Apr 05 2020

%Y Cf. A006530, A076271.

%K nonn

%O 1,5

%A _Michael C. Case_, Jan 08 2020