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A016090
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a(n) = 16^(5^n) mod 10^n: Automorphic numbers ending in digit 6, with repetitions.
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32
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6, 76, 376, 9376, 9376, 109376, 7109376, 87109376, 787109376, 1787109376, 81787109376, 81787109376, 81787109376, 40081787109376, 740081787109376, 3740081787109376, 43740081787109376, 743740081787109376, 7743740081787109376, 7743740081787109376
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OFFSET
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1,1
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COMMENTS
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Also called congruent numbers.
a(n)^2 == a(n) (mod 10^n), that is, a(n) is idempotent of Z[10^n].
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 2^n and a(n) - 1 is divisible by 5^n. - Eric M. Schmidt, Aug 18 2012
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REFERENCES
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R. Cuculière, Jeux Mathématiques, in Pour la Science, No. 6 (1986), 10-15.
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.
A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see pp. 63, 419.
C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
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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) = 16^(5^n) mod 10^n.
a(2*n) = (3*a(n)^2 - 2*a(n)^3) mod 10^(2*n). - Sylvie Gaudel, Mar 12 2018
a(n) = 2^(10^n) mod 10^n for n >= 2. - Peter Bala, Nov 10 2022
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EXAMPLE
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a(5) = 09376 because 09376^2 == 87909376 ends in 09376.
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MAPLE
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MATHEMATICA
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PROG
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(Sage) [crt(0, 1, 2^n, 5^n) for n in range(1, 1001)] # Eric M. Schmidt, Aug 18 2012
(Magma) [Modexp(16, 5^n, 10^n): n in [1..30]]; // Bruno Berselli, Mar 13 2018
(GAP) List([1..22], n->PowerModInt(16, 5^n, 10^n)); # Muniru A Asiru, Mar 20 2018
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CROSSREFS
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A018248 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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