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A108394 Least k for which f(k) = (1 + f(0)^n + f(1)^n + ... + f(k-1)^n)/k, f(0) = 1, is nonintegral.

%I A108394 #72 Feb 26 2025 01:59:20
%S A108394 43,89,97,214,19,239,37,79,83,239,31,431,19,79,23,827,43,173,31,103,
%T A108394 94,73,19,243,141,101,53,811,47,1077,19,251,29,311,134,71,23,86,43,47,
%U A108394 19,419,31,191,83,337,59,1559,19,127,109,163,67,353,83,191,83,107,19,503
%N A108394 Least k for which f(k) = (1 + f(0)^n + f(1)^n + ... + f(k-1)^n)/k, f(0) = 1, is nonintegral.
%C A108394 a(n) is known to be finite for n <= 10^14 (Kobayashi and Seki). - _Stan Wagon_, Dec 13 2024
%D A108394 Ian Stewart, Professor Stewart's Hoard of Mathematical Treasures, "Life, Recursion and Everything", Basic Books, NY, 2009, p. 239-240.
%H A108394 Marshall Buck, Mark Motley, and Stan Wagon, <a href="/A108394/b108394.txt">Table of n, a(n) for n = 2..100000</a> [Computed using the Mathematica code in the following link. The unusually large b-file is included with the OEIS editors' permission.]
%H A108394 Marshall Buck, Mark Motley, and Stan Wagon, <a href="/A108394/a108394_1.txt">Mathematica code to compute A108394</a>
%H A108394 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H A108394 Yuh Kobayashi and Shin-ichiro Seki, <a href="https://doi.org/10.48550/arXiv.2410.23240">A note on non-integrality of the (k,l)-Göbel sequences</a>, arXiv:2410.23240 [math.NT], 2023.
%H A108394 Yuh Kobayashi and Shin-ichiro Seki, <a href="https://arxiv.org/abs/2502.17448">On the length over which k-Göbel sequences remain integers</a>, arXiv:2502.17448 [math.CO], 2025.
%H A108394 Rinnosuke Matsuhira, Toshiki Matsusaka, and Koki Tsuchida, <a href="https://doi.org/10.48550/arXiv.2307.09741">How long can k-Göbel sequences remain integers?</a>, arXiv:2307.09741 [math.NT], 2023.
%H A108394 Alex Stone, <a href="https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/">The Astonishing Behavior of Recursive Sequences</a>, Quanta Magazine, Nov 16 2023, 13 pages.
%F A108394 Matsuhira, Matsusaka, & Tsuchida prove that a(n) >= 19 and a(n) ≠ 41. - _Charles R Greathouse IV_, Nov 17 2023
%p A108394 See link.
%t A108394 primes = DeleteCases[Prime[Range[9, PrimePi[11000]]], 41];
%t A108394 yModPrime[p_, k_] := (i = 1; Nest[(i++;
%t A108394       Mod[# + PowerMod[(# ModularInverse[i - 1, p]), k, p], p]) &, 2, p - 1]);
%t A108394 cGen[k_ /; MemberQ[{6, 14}, Mod[k, 18]], _] := 19;
%t A108394 cGen[k_, M_] := Module[{x = 2, L = M!, n},
%t A108394    Do[x = Mod[(n - 1) x + PowerMod[x, k, L], L]; L /= n;
%t A108394     If[Divisible[x, n], x /= n, Return[n, Module]], {n, 2, M}]; ∞];
%t A108394 cBound[k_, start_ : Automatic] := If[MemberQ[{6, 14}, Mod[k, 18]], 19,
%t A108394   SelectFirst[If[IntegerQ[start], Select[primes, # ≥ start &], primes],
%t A108394    yModPrime[#, k] != 0 &]];
%t A108394 c[k_, start_ : Automatic] := cGen[k, cBound[k, start]];
%t A108394 c /@ Range[2, 10] (* Marshall Buck, Mark Motley, and _Stan Wagon_, Dec 13 2024 *)
%Y A108394 Cf. A003504, A005166, A005167.
%Y A108394 First column of A097398.
%K A108394 nonn
%O A108394 2,1
%A A108394 _William Rex Marshall_, Jul 02 2005