A290788 Values of n such that 6^n ends in n, or expomorphic numbers in "base" 6.

6, 56, 656, 8656, 38656, 238656, 7238656, 47238656, 447238656, 7447238656, 27447238656, 227447238656, 3227447238656

Offset

1,1

Comments

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.]

Links

Table of n, a(n) for n=1..13.

Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.

Example

6^6 = 46656 ends in 6, so 6 is a term.
6^56 = ...656 ends in 56, so 56 is another term.

Mathematica

Select[Range[10^6], PowerMod[6, #, 10^(1 + Floor@ Log10[#])] == # &] (* Michael De Vlieger, Apr 13 2021 *)

Prog

(PARI) is(n)=my(m=10^#digits(n)); Mod(6, m)^n==n \\ Charles R Greathouse IV, Aug 10 2017

Crossrefs

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A289138, A306570 (base 5), A306686 (base 9).

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

Sequence in context: A285166 A182955 A053336 * A215507 A112699 A093197

Adjacent sequences: A290785 A290786 A290787 * A290789 A290790 A290791

Keyword

nonn,base,more

Author

Bernard Schott, Aug 10 2017

Extensions

a(6)-a(9) from Charles R Greathouse IV, Aug 10 2017

a(10)-a(13) from Chai Wah Wu, Apr 13 2021

Status

approved