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

1,5

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.

Links

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

Formula

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

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

Example

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.

Prog

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

Crossrefs

Cf. A006530, A076271.

Sequence in context: A351558 A226240 A109468 * A319690 A341419 A366503

Adjacent sequences: A331029 A331030 A331031 * A331033 A331034 A331035

Keyword

nonn

Author

Michael C. Case, Jan 08 2020

Status

approved