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A288845 Values of n such that 4^n ends in n, or expomorphic numbers in base 4. 6


%S 6,96,896,8896,28896,728896,1728896,11728896,411728896,90411728896,

%T 290411728896,5290411728896,55290411728896,555290411728896,

%U 2555290411728896,302555290411728896,2302555290411728896,22302555290411728896,622302555290411728896,3622302555290411728896

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

%C 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.]

%C 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_

%H Charles W. Trigg, <a href="https://cms.math.ca/crux/backfile/Crux_v7n06_Jun.pdf">Problem 559</a>, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.

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

%o (PARI) isok(n) = 10^valuation(4^n-n, 10)>n; \\ after A003226; _Michel Marcus_, Jun 18 2017

%o (PARI) is(n)=my(m=10^#digits(n)); Mod(4, m)^n==n \\ _Charles R Greathouse IV_, Aug 10 2017

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

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

%K nonn,base

%O 1,1

%A _Bernard Schott_, Jun 18 2017

%E a(6)-a(9) from _Gheorghe Coserea_, Jun 21 2017

%E a(10)-a(11) from _Robert G. Wilson v_, Jun 24 2017

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Last modified April 23 02:43 EDT 2019. Contains 322380 sequences. (Running on oeis4.)