A108394 -id:A108394 - 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

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 *)

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

A288676 Numbers k such that A108394(k) is not a prime.

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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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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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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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A378851 Values of p for which x_p = s_p/p in the modified Göbel sequence is nonintegral.

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