A003504 -id:A003504 - cp's OEIS frontend

A115632 Decimal expansion of asymptotic constant in Goebel's sequence A003504.

1, 0, 4, 7, 8, 3, 1, 4, 4, 7, 5, 7, 6, 4, 1, 1, 2, 2, 9, 5, 5, 9, 9, 0, 9, 4, 6, 2, 7, 4, 3, 1, 3, 7, 5, 5, 4, 5, 9, 0, 5, 8, 7, 6, 1, 2, 8, 6, 0, 2, 3, 3, 0, 9, 6, 9, 5, 1, 0, 4, 0, 6, 4, 8, 5, 3, 5, 3, 6, 0, 5, 9, 0, 4, 9, 7, 2, 6, 2, 3, 1, 7, 9, 7, 5, 1, 3, 0, 9, 7, 9, 0, 0, 0, 7, 0, 9, 9, 4, 7, 9, 5, 1, 1, 3

Offset: 1

Eric W. Weisstein, Jan 27 2006

			1.04783144757641122955990946274313755459058761286023309695104064853536...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.10, p. 446.

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.
Eric Weisstein's World of Mathematics, Goebel's Sequence.

Cf. A003504.

{a(n)=local(t=log(2)/2); for(k=2, 14, t+= (log(1+(k-1)/exp(2^(k-1)*t))-log(k))/2^k); t=exp(t-suminf(k=15, log(k)/2^k)); floor(t*10^(n-1))%10} /* Michael Somos, Apr 02 2006 */

A259878 Gobel's sequence A003504 read mod 43.

Original entry on oeis.org

1, 2, 3, 5, 10, 28, 25, 37, 10, 20, 15, 38, 19, 42, 36, 34, 2, 35, 39, 31, 13, 2, 6, 26, 28, 29, 4, 14, 42, 5, 20, 17, 4, 20, 16, 29, 42, 13, 42, 20, 8, 23, 33

Offset: 0

N. J. A. Sloane, Jul 10 2015

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]

Cf. A003504

t1:=[1,2]; a:=2;
for n from 1 to 41 do
b:=a*(a+n)/(n+1) mod 43;
t1:=[op(t1),b]; a:=b; od:
t1;

A116603 Coefficients in asymptotic expansion of sequence A052129.

Original entry on oeis.org

1, 2, -1, 4, -21, 138, -1091, 10088, -106918, 1279220, -17070418, 251560472, -4059954946, 71250808916, -1351381762990, 27552372478592, -601021307680207, 13969016314470386, -344653640328891233, 8997206549370634644, -247772400254700937149, 7178881153198162073002

Offset: 0

Michael Somos, Feb 18 2006

sign

			G.f. = 1 + 2*x - x^2 + 4*x^3 - 21*x^4 + 138*x^5 - 1091*x^6 + 10088*x^7 + ...

Steven R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 172 (2017), 145-159.
Chao-Ping Chen and X.-F. Han, On Somos' quadratic recurrence constant, Journal of Number Theory, 166 (2016), 31-40.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article no. 6.
Gergo Nemes, On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant, Applicable Analysis and Discrete Mathematics, 5(1) (2011), 60-66.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
Eric Weisstein's World of Mathematics, Goebel's Sequence.
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.
Aimin Xu, Approximations of the generalized-Euler-constant function and the generalized Somos' quadratic recurrence constant, Journal of Inequalities and Applications, Vol. 2019 (2019), Article No. 198.

Cf. A052129, A112302, A123851, A123852, A123853, A123854.

Cf. A000670, A084785.

terms = 20; A[] = 1; Do[A[x] = -A[x] + 2/A[x/(1+x)]^(-1/2)*(1+x) + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jul 28 2011, updated Jan 12 2018 *)

{a(n) = my(A); if( n<0, 0, A=1; for( k=1, n, A = truncate( A + O(x^k)) + x * O(x^k); A = -A + 2 / subst(A^(-1/2), x, x/(1 + x)) * (1 + x);); polcoeff(A, n))};

a(0) = 1; thereafter, a(n) = (1/n)*Sum_{j=1..n} (-1)^(j-1)*2*b(j)*a(n-j), where b(j) = A000670(j) [Nemes]. - N. J. A. Sloane, Sep 11 2017
G.f. A(x) satisfies (1 + x)^2 = A(x)^2 / A(x/(1 + x)).
A003504(n+1) ~ C^(2^n) * (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...) where C = 1.04783144757... (see A115632).
A052129(n) ~ s^(2^n) / (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...) where s = 1.661687949633... (see A112302).
From Seiichi Manyama, May 26 2025: (Start)
G.f.: Product_{k>=1} (1 + k*x)^(1/2^k).
G.f.: exp(2 * Sum_{k>=1} (-1)^(k-1) * A000670(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A084785. (End)
a(n) ~ (-1)^(n+1) * (n-1)! / log(2)^(n+1). - Vaclav Kotesovec, May 27 2025

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 *)

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

A061315 Array read by antidiagonals: T(n,k)=T(n,k-1)*(T(n,k-1)+k-1)/k with T(n,1)=n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 5, 6, 4, 1, 10, 16, 10, 5, 1, 28, 76, 40, 15, 6, 1, 154, 1216, 430, 85, 21, 7, 1, 3520, 247456, 37324, 1870, 161, 28, 8, 1, 1551880

Offset: 1

Henry Bottomley, Apr 24 2001

Not always an integer

			1,1,1,1,1,1,1,1,1,1,...
2,3,5,10,28,154,3520,1551880,267593772160,7160642690122633501504,...
3,6,16,76,1216,247456,61235956672/7,468730299066952899064/49,...
4,10,40,430,37324,232211266,7703153350084336,7417321441864447837991470393906,
5,15,85,1870,700876,81871778626,957569733696568731376,...
6,21,161,6601,8719921,12672844307641,22942997549397847673832961,...
7,28,280,19810,78503068,1027122012987994,...

Rows include A000012 and A003504. Columns include A000027, A000217, A006007, A061319 and A061321.

a(n) =A061314(n, k-1)/k

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.

A061322 a(n) = a(n-1) * (1 + a(n-1)/n^2) with a(0) = 2.

Original entry on oeis.org

2, 6, 15, 40, 140, 924, 24640, 12415040, 2408343949440, 71606426901226335015040, 51274803735606705472274088614112357905277056, 21728144612603201307908899563300049012978385050783094682272184269369267136230071558272

Offset: 0

Henry Bottomley, Apr 24 2001

nonn

Only first 42 terms are integers (see A003504).

			a(2) = 6 * (1 + 6/2^2) = 15.

Eric Weisstein's World of Mathematics, Goebel's Sequence.

Block[{n = 0}, NestList[#*(1 + #/++n^2) &, 2, 11]] (* Paolo Xausa, Apr 17 2024 *)

{a(n) = local(x); if( n<1, 2 * (n==0), (x = a(n-1)) + (x/n)^2)} /* Michael Somos, Apr 02 2006 */

a(n) = a(n-1) + A003504(n+1)^2, a(n-1) = n * A003504(n+1). a(n) = A061314(2, n).

cp's OEIS Frontend

A115632 Decimal expansion of asymptotic constant in Goebel's sequence A003504.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

PARI

A259878 Gobel's sequence A003504 read mod 43.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A116603 Coefficients in asymptotic expansion of sequence A052129.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

References

Links

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A061315 Array read by antidiagonals: T(n,k)=T(n,k-1)*(T(n,k-1)+k-1)/k with T(n,1)=n.

Views

Author

Keywords

Comments

Examples

Crossrefs