A288845 - OEIS

A288845

Values of n such that 4^n ends in n, or expomorphic numbers in base 4.

6, 96, 896, 8896, 28896, 728896, 1728896, 11728896, 411728896, 90411728896, 290411728896, 5290411728896, 55290411728896, 555290411728896, 2555290411728896, 302555290411728896, 2302555290411728896, 22302555290411728896, 622302555290411728896, 3622302555290411728896

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OFFSET

1,1

COMMENTS

Definition: For positive integers b (as base) and n, the positive integer (allowing initial zeros) a(n) is expomorphic relative to base b (here 4) 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.]

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, "0411728896" would be included because the last 10 digits of 4^0411728896 are 0411728896, and also 02555290411728896" because the last 17 digits of 4^02555290411728896 are "02555290411728896". However, these are not in the sequence as defined here. - Jon E. Schoenfield

LINKS

Table of n, a(n) for n=1..20.

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

EXAMPLE

4^6 = 4096 ends in 6, so 6 is a term; 4^96 = ....896 ends in 96, so 96 is another term.

PROG

(PARI) isok(n) = 10^valuation(4^n-n, 10)>n; \\ after A003226; Michel Marcus, Jun 18 2017

(PARI) is(n)=my(m=10^#digits(n)); Mod(4, m)^n==n \\ Charles R Greathouse IV, Aug 10 2017

CROSSREFS

Cf. A064541 (base 2), A183613 (base 3).

Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers), A133614.

Sequence in context: A279868 A055358 A030989 * A006044 A202078 A227262