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A007185
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Automorphic numbers ending in digit 5: a(n) = 5^(2^n) mod 10^n.
(Formerly M3940)
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35
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5, 25, 625, 625, 90625, 890625, 2890625, 12890625, 212890625, 8212890625, 18212890625, 918212890625, 9918212890625, 59918212890625, 259918212890625, 6259918212890625, 56259918212890625, 256259918212890625, 2256259918212890625, 92256259918212890625
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OFFSET
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1,1
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COMMENTS
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Conjecture: For any m coprime to 10 and for any k, the density of n such that a(n) == k (mod m) is 1/m. - Eric M. Schmidt, Aug 01 2012
a(n) is the unique positive integer less than 10^n such that a(n) is divisible by 5^n and a(n) - 1 is divisible by 2^n. - Eric M. Schmidt, Aug 18 2012
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REFERENCES
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V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
Ya. I. Perelman, Algebra can be fun, pp. 97-98.
C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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C. P. Schut, Idempotents, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
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FORMULA
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a(n) = 5^(2^n) mod 10^n.
a(n)^2 == a(n) (mod 10^n), that is, a(n) is an idempotent in Z[10^n].
a(2n) = (3*a(n)^2 - 2*a(n)^3) mod 10^(2n). - Sylvie Gaudel, Mar 10 2018
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EXAMPLE
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625 is in the sequence because 625^2 = 390625, which ends in 625.
90625 is in the sequence because 90625^2 = 8212890625, which ends in 90625.
90635 is not in the sequence because 90635^2 = 8214703225, which does not end in 90635.
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MAPLE
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a:= n-> 5&^(2^n) mod 10^n: seq(a(n), n=1..25); # Alois P. Heinz, Mar 11 2018
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MATHEMATICA
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PROG
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(Sage) [crt(1, 0, 2^n, 5^n) for n in range(1, 1001)] # Eric M. Schmidt, Aug 18 2012
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CROSSREFS
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A018247 gives the associated 10-adic number.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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