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

43, 89, 97, 214, 19, 239, 37, 79, 83, 239, 31, 431, 19, 79, 23, 827, 43, 173, 31, 103, 94, 73, 19, 243, 141, 101, 53, 811, 47, 1077, 19, 251, 29, 311, 134, 71, 23, 86, 43, 47, 19, 419, 31, 191, 83, 337, 59, 1559, 19, 127, 109, 163, 67, 353, 83, 191, 83, 107, 19, 503

Offset: 2

Author

William Rex Marshall, Jul 02 2005

Keywords

• nonn

Comments

a(n) is known to be finite for n <= 10^14 (Kobayashi and Seki). - Stan Wagon, Dec 13 2024

References

• Ian Stewart, Professor Stewart's Hoard of Mathematical Treasures, "Life, Recursion and Everything", Basic Books, NY, 2009, p. 239-240.

Links

• Marshall Buck, Mark Motley, and Stan Wagon, Table of n, a(n) for n = 2..100000 [Computed using the Mathematica code in the following link. The unusually large b-file is included with the OEIS editors' permission.]
• Marshall Buck, Mark Motley, and Stan Wagon, Mathematica code to compute A108394
• R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
• Yuh Kobayashi and Shin-ichiro Seki, A note on non-integrality of the (k,l)-Göbel sequences, arXiv:2410.23240 [math.NT], 2023.
• Yuh Kobayashi and Shin-ichiro Seki, On the length over which k-Göbel sequences remain integers, arXiv:2502.17448 [math.CO], 2025.
• Rinnosuke Matsuhira, Toshiki Matsusaka, and Koki Tsuchida, How long can k-Göbel sequences remain integers?, arXiv:2307.09741 [math.NT], 2023.
• Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.

Crossrefs

Cf. A003504, A005166, A005167.

First column of A097398.

Programs

Maple

See link.

Mathematica

primes = DeleteCases[Prime[Range[9, PrimePi[11000]]], 41];
yModPrime[p_, k_] := (i = 1; Nest[(i++;
      Mod[# + PowerMod[(# ModularInverse[i - 1, p]), k, p], p]) &, 2, p - 1]);
cGen[k_ /; MemberQ[{6, 14}, Mod[k, 18]], _] := 19;
cGen[k_, M_] := Module[{x = 2, L = M!, n},
   Do[x = Mod[(n - 1) x + PowerMod[x, k, L], L]; L /= n;
    If[Divisible[x, n], x /= n, Return[n, Module]], {n, 2, M}]; ∞];
cBound[k_, start_ : Automatic] := If[MemberQ[{6, 14}, Mod[k, 18]], 19,
  SelectFirst[If[IntegerQ[start], Select[primes, # ≥ start &], primes],
   yModPrime[#, k] != 0 &]];
c[k_, start_ : Automatic] := cGen[k, cBound[k, start]];
c /@ Range[2, 10] (* Marshall Buck, Mark Motley, and Stan Wagon, Dec 13 2024 *)

Formula

Matsuhira, Matsusaka, & Tsuchida prove that a(n) >= 19 and a(n) ≠ 41. - Charles R Greathouse IV, Nov 17 2023