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A306686 Values of n such that 9^n ends in n, or expomorphic numbers relative to "base" 9. 3
9, 89, 289, 5289, 45289, 745289, 2745289, 92745289, 392745289, 7392745289, 97392745289, 597392745289, 7597392745289, 87597392745289, 8087597392745289, 48087597392745289, 748087597392745289, 10748087597392745289, 610748087597392745289, 5610748087597392745289 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Definition: For positive integers b (as base) and n, the positive integer (allowing initial zeros) k(n) is expomorphic relative to base b (here 9) if k(n) has exactly n decimal digits and if b^k(n) == k(n) (mod 10^n) or, equivalently, b^k(n) ends in k(n). [See Crux Mathematicorum link.]

For sequences in the OEIS, no term is allowed to begin with a digit 0 (except for the 1-digit number 0 itself). However, in the problem as defined in the Crux Mathematicorum article, leading 0 digits are allowed; under that definition a(n) = k(n) until the first k(n) which begins with digit 0. When k(n) begins with 0, then, a(n) is the next term of the sequence k(n) which doesn't begin with digit 0.

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).

a(n) is the backward concatenation of A133619(0) through A133619(n-1). So, a(1) = 9, a(2) = 89 and so on, with recognition of the former comments about the OEIS and terms beginning with 0. - Davis Smith, Mar 07 2019

LINKS

Davis Smith, Table of n, a(n) for n = 1..944

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

EXAMPLE

9^9 = 387420489 ends in 9, so 9 is a term; 9^89 = .....289 ends in 89, so 89 is another term.

PROG

(PARI) tetrmod(b, n, m)=my(t=b); for(i=1, n, if(i>1, t=lift(Mod(b, m)^t), t)); t

for(n=1, 21, if(tetrmod(9, n, 10^n)!=tetrmod(9, n-1, 10^(n-1)), print1(tetrmod(9, n, 10^(n-1)), ", "))) \\ Davis Smith, Mar 09 2019

CROSSREFS

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A306570 (base 5), A290788 (base 6), A321970 (base 7), A289138 (smallest expomorphic number in base n).

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

Cf. A133619 (leading digits).

Sequence in context: A147884 A178369 A328492 * A055410 A214616 A175371

Adjacent sequences:  A306683 A306684 A306685 * A306687 A306688 A306689

KEYWORD

nonn,base

AUTHOR

Bernard Schott, Mar 05 2019

EXTENSIONS

a(8)-a(20) from Davis Smith, Mar 07 2019

STATUS

approved

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)