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

Offset: 1

Author

Bernard Schott, Jun 18 2017

Keywords

• nonn
• base

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

Examples

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

Links

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

Crossrefs

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

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

Programs

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

Extensions

a(6)-a(9) from Gheorghe Coserea, Jun 21 2017

a(10)-a(11) from Robert G. Wilson v, Jun 24 2017