A234534 Terms of the cycles reached after iterations of numerator(sigma(n)/n) = A017665(n).

1, 8, 15, 127, 128, 144, 255, 403, 448, 512, 1023, 29127, 47360

Offset

1,2

Comments

If all integers were in A014567, then this sequence would not exist and we would be looking at A216200; but some are in A069059, allowing the trajectories of A017665 to go down.

The term of the sequence correspond to the 5 cycles: [1], [15, 8], [448, 127, 128, 255, 144, 403], [1023, 512], [47360, 29127].

Are there some starting x's whose fate will remain unknown, like 276 for A098007?

Are there other cycles to be found?

No other cycles found with largest member less than 10^9.

There are no other cycles with the smallest member < 10^11. All numbers < 10^11 reach one of the five known cycles. - Donovan Johnson, Jan 07 2014

Links

Table of n, a(n) for n=1..13.

Example

Obviously 1 is a fixed point for A017665, so 1 is in the sequence.
A017665(8) = 15 and A017665(15) = 8, so both 8 and 15 are in the sequence.

Prog

(PARI) iscycle(v, nextn) = {for (i=1, #v, if (v[i] == nextn, return (1); ); ); return (0); }
fcycle(n, known) = {v = vector(1); v[1] = n; first = n; while ((nextn = numerator(sigma(n)/n)) <= first, if (vecsearch(known, nextn), return([])); if (iscycle(v, nextn), return (v)); v = concat(v, nextn); n = nextn; ); return ([]); }
fcycles(na, nb) = {known = []; known = [1, 8, 127, 512, 29127]; for (n = na, nb, v = fcycle(n, known); if (#v, print(v, ", "); return(); ); ); } \\ use empty vector for known to search for cycles from start; when a new cycle is found, insert its smallest term to vector known.

Crossrefs

Cf. A000203, A017665.

Sequence in context: A132374 A346314 A361710 * A067686 A283821 A145219

Adjacent sequences: A234531 A234532 A234533 * A234535 A234536 A234537

Keyword

nonn,more

Author

Michel Marcus, Dec 27 2013

Extensions

Missing terms 512 and 1023 noticed by Donovan Johnson added by Michel Marcus, Jan 02 2014

Status

approved