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Numbers m such that the decimal representation of 8^m ends in m.
0

%I #53 Feb 11 2022 09:36:02

%S 56,856,5856,25856,225856,5225856,95225856,895225856,6895225856,

%T 16895225856,416895225856,5416895225856,35416895225856,

%U 7035416895225856,77035416895225856,577035416895225856,1577035416895225856,21577035416895225856,521577035416895225856,1521577035416895225856,81521577035416895225856

%N Numbers m such that the decimal representation of 8^m ends in m.

%C The Crux Mathematicorum link calls these numbers "expomorphic" relative to "base" b, with here b = 8.

%C Under that definition, the term after a(13) = 35416895225856 is not "035416895225856" or "35416895225856" but a(14) = 7035416895225856.

%C Conjecture: if k(n) is "expomorphic" relative to "base" b, then the next one in the sequence, k(n+1), consists of the last n+1 digits of b^k(n).

%C This conjecture is true. See A133618. - _David A. Corneth_, Feb 10 2022

%H Charles W. Trigg, <a href="https://cms.math.ca/wp-content/uploads/crux-pdfs/Crux_v7n06_Jun.pdf">Problem 559</a>, Crux Mathematicorum, pp. 192-194, Vol. 7, Jun. 1981.

%e 8^56 = 374144419156711147060143317175368453031918731001856, so 56 is a term.

%e 8^856 = ...5856 ends in 856, so 856 is another term.

%Y Cf. A064541, A183613, A288845, A306570, A290788, A321970, this sequence, A306686, A289138.

%Y Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).

%Y Cf. A133618 (leading digits).

%K nonn,base

%O 1,1

%A _Bernard Schott_, Feb 10 2022

%E a(7)-a(8) from _Michel Marcus_, Feb 10 2022

%E More terms from _David A. Corneth_, Feb 10 2022