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Numbers n with property that n cycles to itself after sufficiently many iterations of "reverse decimal digits of (n+4)".
5

%I #23 Apr 09 2022 06:15:10

%S 1,2,5,6,9,13,16,24,27,31,35,38,53,57,68,71,75,79,82,93,97,101,122,

%T 137,141,177,181,217,304,319,323,359,363,399,501,505,526,541,545,581,

%U 585,621,708,723,727,763,767,803,905,909,945,949,985,989,1011,1013,1015

%N Numbers n with property that n cycles to itself after sufficiently many iterations of "reverse decimal digits of (n+4)".

%C _Sean A. Irvine_ noted on Mar 14, 2022, that the sequence had more than the previously described 54 terms less than 1000. Specifically, the combined and sorted 22 length-90 iterations for odd starting integers from 1011 to 1053 follow. In fact, it appears that the 22 length-[2*10^(n+1)-110] iterations for odd starting integers from 10^(2*n+1)+11 to 10^(2*n+1)+53 are, for positive n, all terms. - _Hans Havermann_, Mar 18 2022

%C There are no 5-digit terms. Looking through 6-digit integers one finds that, in addition to the above-mentioned 22 length-1890 cycles, there are 449 length-450 cycles (224 odd starting integers from 100101 to 100547, 113 alternate-odd starting integers from 100551 to 100999, and 112 alternate-odd starting integers from 110111 to 110555) and 2 length-225 cycles (for starting integers 100549 and 110559). Added to the existing 2034-term b-file makes for 246114 terms less than one million. Beyond one million one must contend with potentially long runs before entering a cycle. For example, 1001011 requires 25037505 iterations to reach the term 100021. - _Hans Havermann_, Apr 08 2022

%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 15.

%H Hans Havermann, <a href="/A055198/b055198.txt">Table of n, a(n) for n = 1..2034</a>

%o (Python)

%o from sympy import cycle_length

%o f=lambda x:int(''.join(reversed(str(x+4))))

%o def is_A055198(n):

%o return n in cycle_length(f,n,values=True)

%o print([n for n in range(1016) if is_A055198(n)]) # _Pontus von Brömssen_, Mar 19 2022

%Y Cf. A003608, A016081, A016082, A119031.

%K nonn,base

%O 1,2

%A _Henry Bottomley_, Jun 30 2000

%E Edited by _N. J. A. Sloane_, Aug 02 2009

%E Edited by _Charles R Greathouse IV_, Aug 04 2010

%E Deleted two versions of an incorrect conjecture. - _N. J. A. Sloane_, Mar 18 2022

%E Edited by _Hans Havermann_, Mar 18 2022