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A337392
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Minimum m such that the convergence speed of m^^m is equal to n >= 2, where A317905(n) represents the convergence speed of m^^m (and m = A067251(n), the n-th non-multiple of 10).
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4
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5, 25, 15, 95, 65, 385, 255, 1535, 1025, 6145, 4095, 24575, 16385, 98305, 65535, 393215, 262145, 1572865, 1048575, 6291455, 4194305, 25165825, 16777215, 100663295, 67108865, 402653185, 268435455, 1610612735, 1073741825, 6442450945, 4294967295, 25769803775
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OFFSET
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2,1
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COMMENTS
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This sequence has an unbounded number of terms, since it has been proved that the congruence speed (aka "convergence speed") of m^^m (an integer number by definition) covers any value from zero (iff m = 1) to infinity. In particular, for any n >= 2, a(n) == 5 (mod 10).
Moreover, given any m which is congruent to 5 (mod 10), the congruence speed of m corresponds to the 2-adic valuation of (m^2 - 1) minus 1 (e.g., the congruence speed of 15 is equal to 4 since (15^2 - 1) is divisible by 2 exactly 5 times, so that 5 - 1 = 4 = congruence speed of the tetration base 15).
The aforementioned result, let us easily calculate the exact number of stable digits (#S(m, b)) of any tetration m^^b (i.e., the number of its last "frozen" digits) such that m is congruent to 5 (mod 10), for any b >= 3, as follows:
Let k = 1, 2, 3, ...
If m = 20*k - 5, then #S(m, b > 2) = b*(v_2(m^2 - 1) - 1) + 1;
If m = 20*k + 5, then #S(m, b > 2) = (b + 1)*(v_2(m^2 - 1) - 1);
If m = 5, then #S(m, 1) = 1, #S(m, 2) = 3, #S(m, 3) = 4, #S(m, b > 3) = 2.
(End)
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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) = 2^n*(2*cos(Pi*(n-1)/2) - 4*sin(Pi*(n-1)/2) + 5) + 1 iff n == {2,3} (mod 4), 2^n*(-2*cos(Pi*(n-1)/2) + 4*sin(Pi*(n-1)/2) + 5) - 1) iff n == {0,1} (mod 4), for any n >= 2.
O.g.f.: 5*x^2*(1 + 5*x + 4*x^3)/((1 - 2*x)*(1 + 2*x)*(1 + x^2)).
a(n) = (2 - (-1)^n)*2^n + i^((n+1)*(n+2)), with i = sqrt(-1). (End)
n = v_2(a(n)^2 - 1) - 1, where v_2(x) indicates the 2-adic valuation of x. (End)
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EXAMPLE
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For n = 4, a(4) = 15 by Corollary 1 of "https://doi.org/10.7546/nntdm.2021.27.4.43-61" (see Equation 20). - Marco Ripà, Dec 19 2021
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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