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

Author

Michel Marcus, Dec 27 2013

Keywords

• nonn
• more

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

Examples

			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.
		

Crossrefs

Cf. A000203, A017665.

Programs

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.

Extensions

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