A331032 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.

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

Offset: 1

Author

Michael C. Case, Jan 08 2020

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006530, A076271.

Programs

PARI

gpf(n) = if (n==1, 1, my (f=factor(n)); f[#f~, 1])
a(n) = { my (o=1); for (k=0, oo, while (oRémy Sigrist, Apr 05 2020

Formula

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

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