A339313 Smallest prime numbers characterized by a convergence speed of n, assuming a(1) = 2 (since 2^2 <> 2^2^2 (mod 10) and 2^2^2 == 2^2^2^2 (mod 10)).

2, 5, 193, 1249, 22943, 2218751, 4218751, 74218751, 574218751, 30000000001, 281907922943, 581907922943, 6581907922943, 123418092077057, 480163574218751, 19523418092077057, 40476581907922943, 2152996418333704193, 23640476581907922943, 3640476581907922943

Offset

1,1

Comments

It is possible to prove that for any integer n >= 1 there are infinitely many prime numbers with a convergence speed equal to n (invoking Dirichlet's theorem on arithmetic progressions and considering the bases of the form 10^j - 1 + (2*k)*10^j = (2*k + 1)*10^j - 1, since their convergence speed is always equal to j and 10 never divides (2*k + 1)).

Since the only base with a convergence speed of 0 is a = 1 (and 1 is not a prime number), this sequence starts from a(1) = 2, while the convergence speed of 2 has been assumed to be 1 because the tetration 2^^b "freezes" one more rightmost digit for any unitary increment of b for any b >= 3 (the "constant" convergence speed of 2 is 1, even if V(2) = 0 according to the definition used in A317905). In general, a sufficient but not necessary condition to find the constant convergence speed of the base a, is to assume b >= a + 1 (e.g., V(2) corresponds to the new rightmost frozen digit going from 2^^(b >= 3) to 2^^(b + 1)).

This is not a strictly increasing sequence, since 3640476581907922943 = a(20) < a(19) = 23640476581907922943 (while a(19) < a(21) = 803640476581907922943).

For any n >= 3, a(n) == {1,3,7,9}(mod 10), since any prime above 5 is coprime to 10.

Links

Table of n, a(n) for n=1..20.

Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, 2020, 26(3), 245-260.

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.

Example

For n = 3, a(3) = 193, since 193 is the smallest prime number which is characterized by a convergence speed of 3.

Crossrefs

Cf. A000040, A063006, A091661, A290372, A290373, A290374, A290375, A317905, A337833.

Sequence in context: A319143 A100366 A387664 * A012975 A012954 A006271

Adjacent sequences: A339310 A339311 A339312 * A339314 A339315 A339316

Keyword

nonn,hard

Author

Marco Ripà, Nov 29 2020

Status

approved