A349827 -id:A349827

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A349826

Iterate x -> A349824(x) starting at n; if trajectory reaches a fixed point then that is a(n), if it ends in the loop (28,33) then a(n) = 28, otherwise a(n) = -1.

+10
3

0, 0, 2, 3, 28, 5, 28, 7, 28, 27, 28, 11, 27, 13, 28, 28, 28, 17, 28, 19, 27, 27, 30, 23, 28, 27, 30, 27, 28, 29, 30, 31, 28, 28, 28, 28, 28, 37, 28, 28, 28, 41, 28, 43, 28, 28, 28, 47, 28, 28, 28, 28, 28, 53, 28, 28, 28, 28, 28, 59, 28, 61, 28, 28, 28, 28, 28

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OFFSET

0,3

COMMENTS

It is conjectured that every trajectory eventually reaches one of the fixed points {primes union 0, 27, 30} or the loop (28, 33).

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..10000

EXAMPLE

Trajectory of 16 is 16, 32, 50, 36, 40, 44, 45, 33, 28, 33, 28, 33, 28, 33, 28, 33, 28, 33, 28, ..., ending at the loop (28, 33), so a(n) = 28.

PROG

(PARI) a(n) = { for (k=0, oo, my (m=if (n==0, 0, my (f=factor(n)); bigomega(f)*sum(k=1, #f~, f[k, 1]*f[k, 2]))); if (n==28 || m==n, return (n), n=m) ) } \\ Rémy Sigrist, Jan 02 2022

CROSSREFS

Cf. A349824-A349827.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 01 2022

EXTENSIONS

More terms from Rémy Sigrist, Jan 02 2022

STATUS

approved

A349825

Number of steps when x -> A349824(x) is iterated starting at n needed to reach fixed point or 28, or -1 if trajectory increases for ever or ends in a nontrivial loop other than (28,33).

+10
1

0, 1, 0, 0, 9, 0, 10, 0, 8, 4, 9, 0, 3, 0, 8, 9, 8, 0, 7, 0, 1, 2, 2, 0, 6, 2, 1, 0, 0, 0, 0, 0, 7, 1, 8, 7, 5, 0, 7, 8, 4, 0, 6, 0, 3, 2, 7, 0, 9, 1, 6, 5, 6, 0, 4, 8, 7, 4, 12, 0, 10, 0, 11, 9, 12, 6, 10, 0, 10, 7, 7, 0, 11, 0, 6, 9, 8, 6, 5, 0, 7, 10, 9, 0