A328865 The first repeating term in the trajectory of n under iterations of A329623, or -1 if no such terms exists.

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

Author

Scott R. Shannon, Nov 24 2019

Keywords

• sign
• base

Comments

A329623(n) = |n - A053392(n)|, where A053392 is the concatenation of the sums of pairs of consecutive digits.

This is the first number which appears twice for the iteration of A329623 starting with n. See A329624 for an explanation of the sequence, and for the number of iterations required before reaching this value. Terms a(9) to a(127) are all 9's, after which the sequence shows large jumps in value, e.g., a(1673) = 62626262626262626262626262637.

All -1 are so far conjectural, see A329624 and A329917 for more information.

The terms > 0 of this sequence are elements of cycles for A329623. Only 2-cycles and fixed points 1, 2, ..., 9 and 4...455 are known. Therefore a(n) is the earliest A329623^{k}(n) = A329623^{k+2}(n) if such k exist. See A328142 for the list of all possible values and more precise definitions. - M. F. Hasler, Dec 06 2019

The first escape value is a(1373) = -1. - Georg Fischer, Jul 16 2020

Examples

			a(10) = 9 as A329623(10) = 9, and A329623(9) = 9, thus 9 is the first repeating value.
a(128) = 182, as A329623(128) = 182, A329623(182) = 728, A329623(728) = 182, thus 182 is the first repeating value.
a(210) = 637, as A329623(210) = 179, A329623(179) = 637, A329623(637) = 273, A329623(273) = 637, thus 637 is the first repeating value.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A329623, A329624, A329917, A329200, A329197.

Programs

PARI

A328865(n,L=n^10)={my(o);-!until(n>L,o==(n=A329623(o=n))&&return(n))} \\ M. F. Hasler, Dec 06 2019

Extensions

Incorrect comment, link and program deleted following an observation by Scott R. Shannon, Nov 27 2019