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.

User: Stuart Clary

Stuart Clary's wiki page.

Stuart Clary has authored 100 sequences. Here are the ten most recent ones:

A234715 Denominator of sum_{k=1..n} 1/(k*H(k)) where H(k) is the harmonic number H(k) = sum_{j=1..n} 1/j.

Original entry on oeis.org

1, 1, 3, 33, 825, 113025, 5538225, 60920475, 46360481475, 330503872435275, 20160736218551775, 1687675389591187637025, 145175524688023551724527525, 166370135063802174111446471957325, 194941377468714112878127508925972294225, 8038017817167489016303831575544615607779425
Offset: 0

Views

Author

Stuart Clary, Dec 29 2013

Keywords

Comments

The corresponding numerators are in A234714.
A124432(n) = a(n) for 0 <= n <= 53, but A124432(54) = 3 * a(54).

Crossrefs

Cf. A001008, A002805, A000254, A096987, A124432, A234714

Programs

  • Mathematica
    nmax = 54; Table[ Denominator[ Sum[ 1/(k HarmonicNumber[k]), {k, 1, n} ] ], {n, 0, nmax} ]

A234714 Numerator of sum_{k=1..n} 1/(k*H(k)) where H(k) is the harmonic number H(k) = sum_{j=1..n} 1/j.

Original entry on oeis.org

0, 1, 4, 50, 1349, 194713, 9917687, 112451057, 87707471002, 638247495586258, 39621419345255038, 3367553690081394959018, 293578866124447319211215128, 340463591070905769538621961175104, 403214792232827898020426758621769680732, 16787247654077861265551571547714793328259156
Offset: 0

Views

Author

Stuart Clary, Dec 29 2013

Keywords

Comments

The corresponding denominators are in A234715.

Crossrefs

Cf. A001008, A002805, A000254, A096987, A124432, A234715

Programs

  • Mathematica
    nmax = 54; Table[ Numerator[ Sum[ 1/(k HarmonicNumber[k]), {k, 1, n} ] ], {n, 0, nmax} ]

A220990 a(n) = 12^(2n+1) + 6 * 12^n + 1: the right Aurifeuillian factor of 12^(6n+3) + 1.

Original entry on oeis.org

19, 1801, 249697, 35842177, 5159904769, 743009863681, 106993223294977, 15407021789577217, 2218611109320327169, 319479999401581608961, 46005119909741205651457, 6624737266953695061344257, 953962166440743626203987969
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220989.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220989.

Programs

  • Mathematica
    Table[12^(2n+1) + 6 * 12^n + 1, {n, 0, 10}]
LinearRecurrence[{157,-1884,1728},{19,1801,249697},20] (* Harvey P. Dale, Mar 26 2022 *)
  • PARI
    a(n)=12^(2*n+1)+6*12^n+1 \\ Charles R Greathouse IV, Sep 28 2015

Formula

Aurifeuillian factorization: 12^(6n+3) + 1 = (12^(2n+1) + 1) * A220989(n) * a(n).
G.f.: -(2736*x^2-1182*x+19) / ((x-1)*(12*x-1)*(144*x-1)). - Colin Barker, Jan 03 2013

A220989 a(n) = 12^(2n+1) - 6 * 12^n + 1: the left Aurifeuillian factor of 12^(6n+3) + 1.

Original entry on oeis.org

7, 1657, 247969, 35821441, 5159655937, 743006877697, 106993187463169, 15407021359595521, 2218611104160546817, 319479999339664244737, 46005119908998197280769, 6624737266944778960896001, 953962166440636632998608897
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220990.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[12^(2n+1) - 6 * 12^n + 1, {n, 0, 20}]

Formula

Aurifeuillian factorization: 12^(6n+3) + 1 = (12^(2n+1) + 1) * a(n) * A220990(n).
G.f.: -(1008*x^2+558*x+7) / ((x-1)*(12*x-1)*(144*x-1)). [Colin Barker, Jan 03 2013]

A220988 The right Aurifeuillian factor of 11^(22n+11) + 1.

Original entry on oeis.org

407353, 4572972882642803, 109245858982819139102535553, 2812355783638980226466572392952970603, 72895462357781065526518523423275265184080402953, 1890603163831201090586603020695655490130990020251181357603
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220987.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[z = 11^n; 161051 z^10 + 161051 z^9 + 73205 z^8 + 14641 z^7 - 1331 z^6 - 1331 z^5 - 121 z^4 + 121 z^3 + 55 z^2 + 11 z + 1, {n, 0, 10}]

Formula

a(n) = 161051 z^10 + 161051 z^9 + 73205 z^8 + 14641 z^7 - 1331 z^6 - 1331 z^5 - 121 z^4 + 121 z^3 + 55 z^2 + 11 z + 1 with z = 11^n.
Aurifeuillian factorization: 11^(22n+11) + 1 = (11^(2n+1) + 1) * A220987(n) * a(n).

A220987 The left Aurifeuillian factor of 11^(22n+11) + 1.

Original entry on oeis.org

58367, 3812903020530517, 107454987376543082369146967, 2808133028073215608147547774721982717, 72885505321551844061773948114862247606146502767, 1890579685660625069233746109183146734516524279847333062117
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220988.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[z = 11^n; 161051 z^10 - 161051 z^9 + 73205 z^8 - 14641 z^7 - 1331 z^6 + 1331 z^5 - 121 z^4 - 121 z^3 + 55 z^2 - 11 z + 1, {n, 0, 10}]

Formula

a(n) = 161051 z^10 - 161051 z^9 + 73205 z^8 - 14641 z^7 - 1331 z^6 + 1331 z^5 - 121 z^4 - 121 z^3 + 55 z^2 - 11 z + 1 with z = 11^n.
Aurifeuillian factorization: 11^(22n+11) + 1 = (11^(2n+1) + 1) * a(n) * A220988(n).

A220985 The left Aurifeuillian factor of 10^(20n+10) + 1.

Original entry on oeis.org

3541, 904806804901, 99004980069800499001, 9990004998000699800049990001, 999900004999800006999800004999900001, 99999000004999980000069999800000499999000001, 9999990000004999998000000699999800000049999990000001
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220986.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[10^(8n+4) - 10^(7n+4) + 5 * 10^(6n+3) - 2 * 10^(5n+3) + 7 * 10^(4n+2) - 2 * 10^(3n+2) + 5 * 10^(2n+1) - 10^(n+1) + 1, {n, 0, 20}]

Formula

a(n) = 10^(8n+4) - 10^(7n+4) + 5 * 10^(6n+3) - 2 * 10^(5n+3) + 7 * 10^(4n+2) - 2 * 10^(3n+2) + 5 * 10^(2n+1) - 10^(n+1) + 1.
Aurifeuillian factorization: 10^(20n+10) + 1 = (10^(4n+2) + 1) * a(n) * A220986(n).

A220984 The right Aurifeuillian factor of 7^(14n+7) + 1.

Original entry on oeis.org

911, 46489241, 4845303761663, 560176314330212777, 65739735996793498937711, 7731453717973685046293120441, 909551411151743369070229385367263, 107007034358477098527617255914118283977, 12589257482346423369016062830670344414194511
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220983.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[7^(6n+3) + 7^(5n+3) + 3 * 7^(4n+2) + 7^(3n+2) + 3 * 7^(2n+1) + 7^(n+1) + 1, {n, 0, 20}]

Formula

a(n) = 7^(6n+3) + 7^(5n+3) + 3 * 7^(4n+2) + 7^(3n+2) + 3 * 7^(2n+1) + 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * A220983(n) * a(n).
G.f.: -(1483484787696419039*x^6 -1087259214306211086*x^5 +71725962948861585*x^4 -562870083909028*x^3 +609660625665*x^2 -78551886*x +911) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A220983 The left Aurifeuillian factor of 7^(14n+7) + 1.

Original entry on oeis.org

113, 34925927, 4651514210561, 556919483179733591, 65684998500756890925713, 7730533744900130305342957127, 909535949164303794596648514307361, 107006774488854204226839526889653524791, 12589253114717671385404089651370543317211313
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220984.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1, {n, 0, 20}]

Formula

a(n) = 7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * a(n) * A220984(n).
G.f.: -(184010736563880737*x^6 +268740854387875086*x^5 +14564007567924591*x^4 +73553506117028*x^3 +123792021759*x^2 +19415886*x +113) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A220982 a(n) = 6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1: the right Aurifeuillian factor of 6^(12n+6) + 1.

Original entry on oeis.org

97, 55117, 62169337, 78727802257, 101638351073377, 131638631590149697, 170585384377200633217, 221074709452366968135937, 286511970539849391404729857, 371319329591314394530363646977, 481229811357035602199451623479297
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220981.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1, {n, 0, 20}]

Formula

Aurifeuillian factorization: 6^(12n+6) + 1 = (6^(4n+2) + 1) * A220981(n) * a(n).
G.f.: -(162922752*x^4-124050528*x^3+9947772*x^2-95718*x+97) / ((x-1)*(6*x-1)*(36*x-1)*(216*x-1)*(1296*x-1)). [Colin Barker, Jan 03 2013]

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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