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%I #37 Apr 14 2021 22:41:53
%S 6,56,656,8656,38656,238656,7238656,47238656,447238656,7447238656,
%T 27447238656,227447238656,3227447238656
%N Values of n such that 6^n ends in n, or expomorphic numbers in "base" 6.
%C Definition: For positive integers b (as base) and n, the positive integer (allowing initial 0's) a(n) is expomorphic relative to base b (here 6) if a(n) has exactly n decimal digits and if b^a(n) == a(n) (mod 10^n) or, equivalently, b^a(n) ends in a(n). [See Crux Mathematicorum link.]
%H Charles W. Trigg, <a href="https://cms.math.ca/publications/crux/issue/?volume=7&issue=6">Problem 559</a>, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.
%e 6^6 = 46656 ends in 6, so 6 is a term.
%e 6^56 = ...656 ends in 56, so 56 is another term.
%t Select[Range[10^6], PowerMod[6, #, 10^(1 + Floor@ Log10[#])] == # &] (* _Michael De Vlieger_, Apr 13 2021 *)
%o (PARI) is(n)=my(m=10^#digits(n)); Mod(6,m)^n==n \\ _Charles R Greathouse IV_, Aug 10 2017
%Y Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A289138, A306570 (base 5), A306686 (base 9).
%Y Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
%K nonn,base,more
%O 1,1
%A _Bernard Schott_, Aug 10 2017
%E a(6)-a(9) from _Charles R Greathouse IV_, Aug 10 2017
%E a(10)-a(13) from _Chai Wah Wu_, Apr 13 2021