A097398 -id:A097398 - 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 *)

A005167 a(n+1) = (1 + a(0)^4 + ... + a(n)^4 )/(n+1) (not always integral!).

Original entry on oeis.org

1, 2, 9, 2193, 5782218987645, 223567225753623833253893162919867828939456664850241

Offset: 0

N. J. A. Sloane

Terms are integer until n=A097398(3,2)=97.
Guy states that by computing the sequence modulo 97 it is easy to show that a(97) is not integral. - T. D. Noe, Sep 17 2007
The next term -- a(6) -- has 201 digits. - Harvey P. Dale, Nov 20 2018

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..7
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.

Cf. A005166, A003504.

Cf. A108394.

nxt[{n_,a_,t_}]:={n+1,(1+t)/(n+1),t+((1+t)/(n+1))^4}; NestList[nxt,{0,1,1},5][[All,2]] (* Harvey P. Dale, Nov 20 2018 *)

A095397 Modified juggler map: see A095396. Largest value in trajectory of started n under the juggler map of A095396.

Original entry on oeis.org

1, 2, 36, 4, 36, 36, 36, 8, 140, 10, 36, 36, 46, 36, 58, 36, 70, 36, 82, 36, 96, 36, 110, 24, 52214, 26, 140, 140, 156, 140, 172, 32, 2598, 34, 2978, 36, 86818724, 38, 233046, 40, 262, 42, 4710, 44, 5222, 46, 322, 48, 6352, 50, 364, 52, 7554, 54, 8210, 56, 430, 58

Offset: 1

Labos Elemer, Jun 18 2004

nonn

Parallel to A094716.

			n=37: the trajectory is {37, 225, 3375, 196069, 86818724, 196068, 3374, 224, 36, 10, 4, 2, 1}, the peak is a[37]=86818724

Cf. A007320, A094683, A094716, A095396, A097398.

d[x_]:=d[x]=(1-Mod[x, 2])*Floor[N[x^(2/3), 50]] +Mod[x, 2]*Floor[N[x^(3/2), 50]];d[1]=1; fd[x_]:=Delete[FixedPointList[d, x], -1] Table[Max[fd[w]], {w, 1, m}]

A292996 Index of first nonintegral term in the sequence b(1)=2, b(k)=b(k-1)*(b(k-1)+k-1+n)/k (k>=2).

Original entry on oeis.org

43, 3, 7, 5, 3, 7, 59, 3, 5, 7, 3, 23, 7, 3, 19, 17, 3, 73, 5, 3, 19, 13, 3, 5, 17, 3, 7, 17, 3, 59, 7, 3, 17, 5, 3, 43, 17, 3, 5, 19, 3, 17, 29, 3, 7, 11, 3, 7, 5, 3, 31, 7, 3, 5, 7, 3, 11, 23, 3, 83, 13, 3, 41, 5, 3, 7, 17, 3, 5, 47, 3, 19, 7, 3, 23, 7, 3, 19, 5, 3, 23, 31, 3, 5, 43, 3, 7, 17, 3, 7, 19, 3, 17, 5, 3, 17, 7, 3, 5, 13, 3, 93

Offset: 0

Max Alekseyev, Sep 27 2017

nonn

a(0)=A097398(1,2)=43 corresponds to Goebel's sequence A003504.

First composite term is a(101)=93.

Max Alekseyev, Table of n, a(n) for n = 0..10000
M. Alekseyev, Simple recurrence that fails to be integer for the first time at the 44th term, Mathoverflow, 2017.

Cf. A003504, A097398.

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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A005167 a(n+1) = (1 + a(0)^4 + ... + a(n)^4 )/(n+1) (not always integral!).

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A095397 Modified juggler map: see A095396. Largest value in trajectory of started n under the juggler map of A095396.

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A292996 Index of first nonintegral term in the sequence b(1)=2, b(k)=b(k-1)*(b(k-1)+k-1+n)/k (k>=2).

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