A005167 -id:A005167 - cp's OEIS frontend

A003504 a(0)=a(1)=1; thereafter a(n+1) = (1/n)*Sum_{k=0..n} a(k)^2 (a(n) is not always integral!).

1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856

Offset: 0

N. J. A. Sloane, R. K. Guy

The sequence appears with a different offset in some other sources. - Michael Somos, Apr 02 2006
Also known as Göbel's (or Goebel's) Sequence. Asymptotically, a(n) ~ n*C^(2^n) where C=1.0478... (A115632). A more precise asymptotic formula is given in A116603. - M. F. Hasler, Dec 12 2007
Let s(n) = (n-1)*a(n). By considering the p-adic representation of s(n) for primes p=2,3,...,43, one finds that a(44) is the first nonintegral value in this sequence. Furthermore, for n>44, the valuation of s(n) w.r.t. 43 is -2^(n-44), implying that both s(n) and a(n) are nonintegral. - M. F. Hasler and Max Alekseyev, Mar 03 2009
a(44) is approximately 5.4093*10^178485291567. - Hans Havermann, Nov 14 2017.
The fractional part is simply 24/43 (see page 709 of Guy (1988)).
The more precise asymptotic formula is a(n+1) ~ C^(2^n) * (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...). - Michael Somos, Mar 17 2012

			a(3) = (1 * 2 + 2^2) / 2 = 3 given a(2) = 2.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Sect. E15.
Clifford Pickover, A Passion for Mathematics, 2005.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..16
Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 1.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
H. Ibstedt, Some sequences of large integers, Fibonacci Quart. 28 (1990), 200-203.
Yuh Kobayashi and Shin-ichiro Seki, On the length over which k-Göbel sequences remain integers, arXiv:2502.17448 [math.CO], 2025.
H. W. Lenstra, Jr., R. K. Guy, and N. J. A. Sloane, Correspondence, 1975-1978
N. Lygeros and M. Mizony, Study of primality of terms of a_k(n)=(1+(sum from 1 to n-1)(a_k(i)^k))/(n-1)
Rinnosuke Matsuhira, Toshiki Matsusaka, and Koki Tsuchida, How long can k-Göbel sequences remain integers?, arXiv:2307.09741 [math.NT], 2023.
D. Rusin, Law of small numbers [Broken link]
D. Rusin, Law of small numbers [Cached copy]
Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
Eric Weisstein's World of Mathematics, Göbel's Sequence
D. Zagier, Problems posed at the St Andrews Colloquium, 1996
D. Zagier, Solution: Day 5, problem 3

Cf. A005166, A005167, A097398, A108394, A115632, A116603 (asymptotic formula).

a:=2: L:=1,1,a: n:=15: for k to n-2 do a:=a*(a+k)/(k+1): L:=L,a od:L; # Robert FERREOL, Nov 07 2015

a[n_] := a[n] = Sum[a[k]^2, {k, 0, n-1}]/(n-1); a[0] = a[1] = 1; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Feb 06 2013 *)
With[{n = 14}, Nest[Append[#, (#.#)/(Length[#] - 1)] &, {1, 1}, n - 2]] (* Jan Mangaldan, Mar 21 2013 *)

A003504(n,s=2)=if(n-->0,for(k=1,n-1,s+=(s/k)^2);s/n,1) \\ M. F. Hasler, Dec 12 2007

a=2; L=[1,1,a]; n=15
for k in range(1,n-1):
    a=a*(a+k)//(k+1)
    L.append(a)
print(L) # Robert FERREOL, Nov 07 2015

a(n+1) = ((n-1) * a(n) + a(n)^2) / n if n > 1. - Michael Somos, Apr 02 2006

0 = a(n)*(+a(n)*(a(n+1) - a(n+2)) - a(n+1) - a(n+1)^2) +a(n+1)*(a(n+1)^2 - a(n+2)) if n>1. - Michael Somos, Jul 25 2016

a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral - H. W. Lenstra, Jr.
Corrected and extended by M. F. Hasler, Dec 12 2007
Further corrections from Max Alekseyev, Mar 04 2009

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

Original entry on oeis.org

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

William Rex Marshall, Jul 02 2005

nonn

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

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

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.

Cf. A003504, A005166, A005167.

First column of A097398.

Maple
```
See link.
```

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 *)

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

A005166 a(0) = 1; a(n) = (1 + a(0)^3 + ... + a(n-1)^3)/n (not always integral!).

Original entry on oeis.org

1, 2, 5, 45, 22815, 2375152056927, 2233176271342403475345148513527359103

Offset: 0

N. J. A. Sloane

Terms are integers until n=A097398(2,2)=89.

Guy states that by computing the sequence modulo 89 it is easy to show that a(89) is not integral. - T. D. Noe, Sep 17 2007

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..9
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
N. Lygeros & M. Mizony, Study of primality of terms of a_k(n)=(1+(sum from 1 to n-1)(a_k(i)^k))/(n-1) [dead link]
Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
Eric Weisstein's World of Mathematics, Goebel's Sequence.

Cf. A003504, A005167.

Cf. A108394.

a[0]=1; a[n_]:=(1 + Sum[a[k]^3, {k,0,n-1}])/n; Array[a,7,0] (* Stefano Spezia, Oct 13 2024 *)

A097398 Matrix T(m,x(1)), m>=1, x(1)>=2, read by antidiagonals, where T(m,x(1)) gives the position of the first noninteger term in the sequence defined by x(n)=(x(n-1)*(x(n-1)^m+n-1))/n for n>=2 with exponent m and the given starting value x(1).

Original entry on oeis.org

43, 7, 89, 17, 89, 97, 34, 89, 17, 214, 17, 89, 23, 43, 19, 17, 31, 97, 139, 83, 239, 51, 151, 149, 107, 13, 191, 37, 17, 79, 13, 269, 19, 359, 7, 79, 7, 89, 13, 107, 13, 419, 23, 127, 83, 34, 79, 83, 214, 37, 127, 37, 158, 31, 239

Offset: 1

Hugo Pfoertner, Aug 15 2004

The rectangular table (Table 1, page 35) in Ibstedt's book gives the position of the first noninteger term for parameters x1 and m:
  m\x1: 2   3   4   5   6   7   8   9  10  11
 43   7  17  34  17  17  51  17   7  34
 89  89  89  89  31 151  79  89  79 601
 97  17  23  97 149  13  13  83  23  13
214  43 139 107 269 107 214 139 251 107
 19  83  13  19  13  37  13  37 347  19
239 191 359 419 127 127 239 191 239 461
 37   7  23  37  23  37  17  23   7  37
 79 127 158  79 103 103 163 103 163  79
 83  31  41  83  71  83  71  23  41  31
239 389 169 137 239 239 239 239 239 389

			T(1,3)=a(2)=7: x(1)=3, x(2)=x(1)*(x(1)^1+2-1)/n=3*(3+2-1)/2=6, x(3)=6*(6+3-1)/3=16, x(4)=16*(16+4-1)/4=76, x(5)=76*(76+5-1)/5=1216, x(6)=1216*(1216+6-1)/6=247456, x(7)=247456*(247456+7-1)/7=8747993810+2/7; i.e., x(7) is the first noninteger term in the sequence x(n) = x(n-1)*(x(n-1)^1+n-1)/n, n>=2, x(1)=3.

R. K. Guy, Unsolved Problems in Number Theory, E15.
Henry Ibstedt, Mainly natural numbers - a few elementary studies on Smarandache sequences and other number problems, Henry Ibstedt. - Martinsville, Ind.: Bookman, 2003. Chapter IV, Some Sequences of Large Integers, pp. 32-37.

Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 2.
H. Ibstedt, Some sequences of large integers, Fibonacci Quart. 28 (1990), 200-203.
Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
Eric Weisstein's World of Mathematics: Göbel's Sequence

Cf. A003504 for more references and links, A005166, A005167.

m=10 row corrected by Don Reble, Dec 07 2004, who remarks that the versions in the books of Ibstedt and Guy are both wrong

A288641 Define the sequence {b_n(k)} as the solutions of the recursion (k+1) * b_n(k+1) = b_n(k) * (b_n(k)^(n-1) + k) with b_n(0) = 1. a(n) is the least prime p where p * b_n(p) is not 0 mod p.

Original entry on oeis.org

43, 89, 97, 251, 19, 239, 37, 79, 83, 239, 31, 431, 19, 79, 23, 827, 43, 173, 31, 103, 179, 73, 19, 431, 193, 101, 53, 811, 47, 1427, 19, 251, 29, 311, 137, 71, 23, 499, 43, 47, 19, 419, 31, 191, 83, 337, 59, 1559, 19, 127, 109, 163, 67, 353, 83, 191, 83, 107

Offset: 2

Seiichi Manyama, Jun 13 2017

nonn

If A108394(n) is a prime, a(n) = A108394(n).

			(k+1) * b_2(k+1) = b_2(k) * (b_2(k) + k) with b_2(0) = 1.
b_2(1) == 2, b_2(2) == 3, b_2(3) == 5, ... , b_2(42) == 33 mod 43.
So 43 * b_2(43) == b_2(42) * (b_2(42) + 42) == 24 (> 0) mod 43.

Seiichi Manyama, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Goebel's Sequence

Cf. A003504 ({b_2(n+1)}), A005166 ({b_3(n)}), A005167 ({b_4(n)}), A108394, A288676.

cp's OEIS Frontend

A003504 a(0)=a(1)=1; thereafter a(n+1) = (1/n)*Sum_{k=0..n} a(k)^2 (a(n) is not always integral!).

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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.

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A005166 a(0) = 1; a(n) = (1 + a(0)^3 + ... + a(n-1)^3)/n (not always integral!).

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Comments

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Crossrefs

Programs

Mathematica

A097398 Matrix T(m,x(1)), m>=1, x(1)>=2, read by antidiagonals, where T(m,x(1)) gives the position of the first noninteger term in the sequence defined by x(n)=(x(n-1)*(x(n-1)^m+n-1))/n for n>=2 with exponent m and the given starting value x(1).

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Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A288641 Define the sequence {b_n(k)} as the solutions of the recursion (k+1) * b_n(k+1) = b_n(k) * (b_n(k)^(n-1) + k) with b_n(0) = 1. a(n) is the least prime p where p * b_n(p) is not 0 mod p.

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