A294659 Largest number in the orbit of n under iteration of the map A293975: x -> x/2 if even, x + nextprime(x) else.

0, 8, 8, 8, 12, 8, 20, 8, 20, 12, 24, 12, 32, 20, 32, 16, 36, 20, 44, 20, 44, 24, 52, 24, 56, 32, 56, 28, 60, 32, 68, 32, 72, 36, 72, 36, 80, 44, 80, 40, 84, 44, 92, 44, 92, 52, 100, 48, 104, 56, 104, 52, 112, 56, 116, 56, 116, 60, 120, 60, 128, 68, 132, 64, 132, 72, 140, 68, 140, 72, 144, 72, 152, 80

Offset: 0

Author

M. F. Hasler, Nov 06 2017

Keywords

• nonn

Comments

The trajectory under iterations of A293975 seems to end in the cycle 1 -> 3 -> 8 -> 4 -> 2 -> 1, for any positive starting value n. Therefore a(n) >= 8 for all n > 0.

Obviously also a(n) >= n for all numbers, with equality for powers of two 2^k with k >= 3; a(n) >= n + nextprime(n) >= 2n+2 for all odd numbers.

Record values not of the form f(n) = n + nextprime(n) occur for a(1) = 8, a(5) = 12, a(7) = 20, a(11) = 24, a(13) = 32, a(17) = 36, a(19) = 44, a(23) = 52, a(25) = 56, a(29) = 60, a(31) = 68, a(33) = 72, a(37) = 80, a(41) = 84, a(43) = 92, a(47) = 100, ... We see that in most cases, this equals f(f(n)/2). Exceptions are n = 1, 7, 13, 19, 37, 43, 67, 79, 89, 97, ...

Crossrefs

Cf. A293975, A293982 (size of the orbit).

Programs

PARI

A294659(n,S=[n])={while(#S<#S=setunion(S,[n=A293975(n)]),); vecmax(S)}