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A337836
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a(n) is the smallest base of the form 8 + 10*k which is characterized by a convergence speed of n, where A317905(n) represents the convergence speed of m^^m.
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1
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8, 18, 68, 2318, 7318, 1068, 32318, 501068, 7532318, 3626068, 23157318, 120813568, 3538782318, 1097376068, 110960657318, 49925501068, 1880980188568, 355101282318, 53760863001068, 15613890344818, 587818480188568, 2495167113001068
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OFFSET
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1,1
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COMMENTS
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Let n >= 1. For any t == 8 (mod 10), if 5^n divides (t^2 + 1) and 5^(n + 1) does not divide (t^2 + 1), then V(t) = n (where V(t) represents the convergence speed of t). In particular, the aforementioned property holds for any a(n), since a(n) belongs to the residue class 8 modulo 10 for any n. Moreover, 5^n always divides (a(n) + A340345(n)).
In general, any tetration base m = A067251(n) which is congruent to {2,8}(mod 10) is characterized by a convergence speed equal to the 5-adic valuation of m^2 + 1. Similarly, if m is congruent to 4(mod 10), then the convergence speed of m is given by m + 1, whereas if m belongs to the congruence class 6 modulo 10, then its convergence speed is m - 1. Lastly, for any m congruent to 5 modulo 10, the congruence speed exceeds by 1 the 2-adic valuation of m^2 - 1
Moreover, assuming m > 1, m^m is not congruent to m^m^m if and only if m belongs to the congruence class 18 modulo 20 or 2 modulo 20, whereas if m = A067251(n) is not coprime to 10 and is not equal to 5, then the number of new stable digits from m^m^m to m^m^m^m is always equal to the convergence speed of m. The aforementioned statement, in general, is untrue if m is coprime to 10 (see "Number of stable digits of any integer tetration" in the Links section).
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REFERENCES
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Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6
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LINKS
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FORMULA
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a(n) = g(n) + u(n), where g(n) = (-2^5^n (mod 10^n)) (mod 2*5^n) and where u(n) = [0 iff g(n) <> g(n + 1); 2*5^n iff g(n) = g(n + 1)].
a(n) = 5-adic valuation of a(n)^2 + 1. - Marco Ripà, Dec 31 2021
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EXAMPLE
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For n = 3, a(3) = 68 is characterized by a convergence speed of 3, and it is the smallest base such that V(a) = 3. Moreover, 5^3 has to divide a(3) (i.e., a(3)^2+1 = 4625 = 5^3*37 is a multiple of 5^3).
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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