cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A082732 a(1) = 1, a(2) = 3, a(n) = LCM of all the previous terms + 1.

Original entry on oeis.org

1, 3, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693
Offset: 1

Views

Author

Amarnath Murthy, Apr 14 2003

Keywords

Comments

The LCM is in fact the product of all previous terms. From a(5) onwards the terms alternately end in 57 and 93.

Crossrefs

Cf. A000058, A004168, A144743, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

Programs

  • Mathematica
    a[1] = 1; a[2] = 3; a[n_] := Apply[LCM, Table[a[i], {i, 1, n - 1}]] + 1; Table[ a[n], {n, 1, 10}]
c=1.8806785436830780944921917650127503562630617563236301969047995953391479871\
7695395204087358090874194124503892563356447954254847544689332763; Table[c^(2^n),{n,1,6}] or a = {}; k = 4; Do[AppendTo[a, k]; k = k^2 - k + 1, {n, 1, 10}]; a (* Artur Jasinski, Sep 22 2008 *)

Formula

For n>=3, a(n+1) = a(n)^2 - a(n) + 1.
For n>=3, a(n) = A004168(n-3) + 1. - Max Alekseyev, Aug 09 2019
1/3 = Sum_{n=3..oo} 1/a(n) = 1/4 + 1/13 + 1/157 + 1/24493 + ... or 1 = Sum_{n=3..oo} 3/a(n) = 3/4 + 3/13 + 3/157 + 3/24493 + .... If we take segment of length 1 and cut off in each step fragment of maximal length such that numerator of fraction is 3, denominators of such fractions will be successive numbers of this sequence. - Artur Jasinski, Sep 22 2008
a(n+2)=1.8806785436830780944921917650127503562630617563236301969047995953391\
4798717695395204087358090874194124503892563356447954254847544689332763...^(2^n). - Artur Jasinski, Sep 22 2008

Extensions

More terms from Robert G. Wilson v, Apr 15 2003

A078358 Non-oblong numbers: Complement of A002378.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74
Offset: 1

Views

Author

Wolfdieter Lang, Nov 29 2002

Keywords

Comments

The (primitive) period length k(n)=A077427(n) of the (regular) continued fraction of (sqrt(4*a(n)+1)+1)/2 determines whether or not the Diophantine equation (2*x-y)^2 - (1+4*a(n))*y^2 = +4 or -4 is solvable and the approximants of this continued fraction give all solutions. See A077057.
The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003
Infinite series 1/A078358(n) is divergent. Proof: Harmonic series 1/A000027(n) is divergent and can be distributed on two subseries 1/A002378(k+1) and 1/A078358(m). The infinite subseries 1/A002378(k+1) is convergent to 1, so Sum_{n>=1} 1/A078358(n) is divergent. - Artur Jasinski, Sep 28 2008

References

  • O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Links

Crossrefs

a(n)=(A077425(n)-1)/4.
Cf. A049068 (subsequence), A144786.

Programs

  • Haskell
    a078358 n = a078358_list !! (n-1)
a078358_list = filter ((== 0) . a005369) [0..]
-- Reinhard Zumkeller, Jul 04 2014, May 08 2012
  • Mathematica
    Complement[Range[930], Table[n (n + 1), {n, 0, 30}]] (* and *) Table[Ceiling[Sqrt[n]] + n - 1, {n, 900}] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)
  • PARI
    a(n)=sqrtint(n-1)+n \\ Charles R Greathouse IV, Jan 17 2013
  • Python
    from operator import sub
from sympy import integer_nthroot
def A078358(n): return n+sub(*integer_nthroot(n,2)) # Chai Wah Wu, Oct 01 2024

Formula

4*a(n)+1 is not a square number.
a(n) = ceiling(sqrt(n)) + n -1. - Leroy Quet, Jul 06 2007
A005369(a(n)) = 0. - Reinhard Zumkeller, Jul 05 2014

A144780 Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 6.

Original entry on oeis.org

6, 31, 931, 865831, 749662454731, 561993796032558961827631, 315837026779085485103718410756049100028793244531
Offset: 1

Views

Author

Artur Jasinski, Sep 21 2008

Keywords

Links

Crossrefs

Cf. A000058, A082732, A144779, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788, A144804.

Programs

  • Mathematica
    a = {}; k = 6; Do[AppendTo[a, k]; k = k^2 - k + 1, {n, 1, 10}]; a
NestList[#^2-#+1&,6,10] (* Harvey P. Dale, Dec 19 2024 *)

Formula

a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 6.
a(n) ~ c^(2^n) where is c is 2.350117384... (A144804).

Extensions

a(8) moved to b-file by Hugo Pfoertner, Aug 30 2020

A144784 Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 11.

Original entry on oeis.org

11, 111, 12211, 149096311, 22229709804712411, 494159998001727075769152612720511, 244194103625066907517263589918036880566782292998362610615987380611
Offset: 1

Views

Author

Artur Jasinski, Sep 21 2008

Keywords

Comments

For the "exact" formula, compare the Aho-Sloane reference in A000058. - N. J. A. Sloane, Apr 07 2014

Links

Crossrefs

Cf. A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788, A144808.
See A239900 for another version.

Programs

  • Mathematica
    a = {}; r = 11; Do[AppendTo[a, r]; r = r^2 - r + 1, {n, 1, 10}]; a

Formula

a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 11.
a(n) ~ c^(2^n) where c = 3.242214... (see A144808).

A144779 Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 5.

Original entry on oeis.org

5, 21, 421, 176821, 31265489221, 977530816197201697621, 955566496615167328821993756200407115362021, 913107329453384594090655605142589591944556891901674138343716072975722193082773842421
Offset: 1

Views

Author

Artur Jasinski, Sep 21 2008

Keywords

Examples

			a(0) = 4, a(1) = 4+1 = 5, a(2) = 4*5+1 = 21, a(3) = 4*5*21+1 = 421, a(4) = 4*5*21*421+1 = 176821, ... - _Philippe Deléham_, Apr 19 2013

Links

Crossrefs

Cf. A000058, A082732, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

Programs

  • Mathematica
    a = {}; k = 5; Do[AppendTo[a, k]; k = k^2 - k + 1, {n,1,10}]; a (* Artur Jasinski, Sep 21 2008 *)
NestList[#^2-#+1&,5,8] (* Harvey P. Dale, Jan 17 2012 *)

Formula

a(n) = round(2.127995907464107054577351...)^(2^n) = round(A144803^(2^n)). [corrected by Joerg Arndt, Jan 15 2021]
a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 5.

A144782 Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 9.

Original entry on oeis.org

9, 73, 5257, 27630793, 763460694178057, 582872231554839914154126117193, 339740038317718918529575265905277902175236102890836244082057
Offset: 1

Views

Author

Artur Jasinski, Sep 21 2008

Keywords

Links

Crossrefs

Cf. A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

Programs

  • Mathematica
    a = {}; r = 9; Do[AppendTo[a, r]; r = r^2 - r + 1, {n, 1, 10}]; a
NestList[#^2-#+1&,9,10] (* Harvey P. Dale, Aug 31 2014 *)

Formula

a(n) ~ c^(2^n) with c = 2.918012...
a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 9.

A144783 Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 10.

Original entry on oeis.org

10, 91, 8191, 67084291, 4500302031888391, 20252718378218776104731448680491, 410172601707440572557971589875869064610540321970215293555320591, 168241563191450680898537024308131628447885486994777537422995633998657738457104605412468520116391629012196009150161991233268691
Offset: 1

Views

Author

Artur Jasinski, Sep 21 2008

Keywords

References

  • Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342

Crossrefs

Cf. A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

Programs

  • Mathematica
    a = {}; r = 10; Do[AppendTo[a, r]; r = r^2 - r + 1, {n, 1, 10}]; a (* or *)
Table[Round[3.08435104906918990233569320020272148875011089837398848476442237096569188195734783139337492942278549518507672786196650938869338548385641623^(2^n)], {n, 1, 8}] (* Artur Jasinski *)
NestList[#^2-#+1&,10,8] (* Harvey P. Dale, May 07 2017 *)

Formula

a(n) = round((3.08435104906918990233569320020272148875011089837398848476442237096569...)^(2^n)) = round(A144807^(2^n)). [corrected by Joerg Arndt, Jan 15 2021]
a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 10.

A144785 Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 12.

Original entry on oeis.org

12, 133, 17557, 308230693, 95006159799029557, 9026170399758739819525199160586693, 81471752085480849000657595909467634426991447160798281416700808089557
Offset: 1

Views

Author

Artur Jasinski, Sep 21 2008

Keywords

References

  • Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342

Crossrefs

A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788

Programs

  • Mathematica
    a = {}; r = 12; Do[AppendTo[a, r]; r = r^2 - r + 1, {n, 1, 10}]; a or Table[Round[3.39277252592669675143137065018187376847206615308598784654603692312172475924599026837940758013759324881455503678006543568111163817496672898^(2^n)], {n, 1, 8}] (*Artur Jasinski*)
NestList[#^2-#+1&,12,6] (* Harvey P. Dale, Jan 01 2016 *)

Formula

a(n) =3.39277252592669675143137065018187376847206615308598784654603692312172475924599026837940758013759324881455503678006543568111163817496672898^(2^n) a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 11

A144787 Recurrence sequence a(n+1)=a(n)^3-a(n)+1 and a(1)=2.

Original entry on oeis.org

2, 7, 337, 38272417, 56060590716839257663297, 176186654453940966415101758343368831005891099500239113100063334235777
Offset: 1

Views

Author

Artur Jasinski, Sep 22 2008

Keywords

Comments

For constant c=1.240554576397679299452... see A144810.

Crossrefs

Cf. A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

Programs

  • Mathematica
    a = {}; k = 2; Do[AppendTo[a, k]; k = k^3 - k + 1, {n, 1, 8}]; a (*Artur Jasinski*)
NestList[#^3-#+1&,2,5] (* Harvey P. Dale, Jun 24 2013 *)
Showing 1-9 of 9 results.

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