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: Jean-Claude Babois

Jean-Claude Babois's wiki page.

Jean-Claude Babois has authored 7 sequences.

A372069 Value of p^2 + (p-1)^2/2 as p runs through A001359 (the lesser of twin primes).

Original entry on oeis.org

11, 33, 171, 417, 1233, 2481, 5163, 7491, 15201, 17067, 28017, 33153, 47883, 54531, 58017, 77067, 85443, 108273, 118161, 144771, 180267, 262923, 278211, 318321, 406641, 485073, 537603, 570417, 615681, 650763, 980913, 1010241, 1025067, 1100817, 1163361, 1556523, 1593411, 1649553, 1687521
Offset: 1

Views

Author

N. J. A. Sloane, May 25 2024, based on an email from Jean-Claude Babois

Keywords

Links

A362941 Numbers of the form (p+1)*(p+3) where (p,p+2) is a twin prime pair (cf. A001359).

Original entry on oeis.org

24, 48, 168, 360, 960, 1848, 3720, 5328, 10608, 11880, 19320, 22800, 32760, 37248, 39600, 52440, 58080, 73440, 80088, 97968, 121800, 177240, 187488, 214368, 273528, 326040, 361200, 383160, 413448, 436920, 657720, 677328, 687240, 737880, 779688, 1042440, 1067088, 1104600
Offset: 1

Views

Author

N. J. A. Sloane, Sep 10 2023, following a suggestion from Jean-Claude Babois

Keywords

Links

Crossrefs

Cf. A001359, A108604.

Programs

  • Mathematica
    ((# + 1)*(# + 3)) & /@ Select[Prime[Range[200]], PrimeQ[# + 2] &] (* Amiram Eldar, Sep 10 2023 *)

Formula

a(n) = A108604(n) - 1. - Amiram Eldar, Sep 10 2023

A331764 a(n) = ((p-1)^3 - (p-1)^2)/4 where p is the n-th prime.

Original entry on oeis.org

0, 1, 12, 45, 225, 396, 960, 1377, 2541, 5292, 6525, 11340, 15600, 18081, 23805, 34476, 47937, 53100, 70785, 84525, 92016, 117117, 136161, 168432, 218880, 247500, 262701, 294945, 312012, 348096, 496125, 545025, 624240, 652257, 804972, 838125, 943020
Offset: 1

Views

Author

N. J. A. Sloane, Feb 05 2020 following a suggestion from Jean-Claude Babois

Keywords

Links

Programs

  • Maple
    a:= n-> (p-> ((p-1)^3-(p-1)^2)/4)(ithprime(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 05 2020
  • Mathematica
    Table[((Prime[n] - 1)^3 - (Prime[n] - 1)^2)/4, {n, 20}] (* Eric W. Weisstein, Aug 22 2021 *)
Table[((Prime[n] - 2) (Prime[n] - 1)^2)/4, {n, 20}] (* Eric W. Weisstein, Aug 22 2021 *)
Table[Times @@ (Prime[n] - {1, 1, 2})/4, {n, 20}] (* Eric W. Weisstein, Aug 22 2021 *)
Table[Sum[Floor[i j/Prime[n]], {i, Prime[n] - 1}, {j, Prime[n] - 1}], {n, 20}] (* Eric W. Weisstein, Aug 22 2021 *)

Formula

Theorem: a(n) = Sum_{i=1..p-1, j=1..p-1} floor(i*j/p). The proof is based on the formula for p-g-c-d of Marcelo Polezzi. - Jean-Claude Babois
a(n) == 0 (mod 3) for n >= 3. - Hugo Pfoertner, Aug 23 2021

A224224 Numerators of continued fraction convergents to ( sqrt(4*sqrt(3)-3) - 1 )/4.

Original entry on oeis.org

0, 1, 13, 14, 27, 41, 68, 109, 177, 29314, 29491, 58805, 382321, 441126, 823447, 3734914, 26967845, 704898884, 731866729, 3632365800, 4364232529, 12360830858, 16725063387, 45810957632, 62536021019, 170882999670, 233419020689, 637721041048, 3422024225929, 17747842170693
Offset: 1

Views

Author

Jean-Claude Babois, Apr 08 2013

Keywords

Links

Crossrefs

Cf. A214367.

Programs

  • Mathematica
    Numerator[Convergents[(Sqrt[4 Sqrt[3] - 3] - 1)/4, 50]]

Extensions

Edited by N. J. A. Sloane, Apr 08 2013 at the request of Jean-Claude Babois
Mathematica code adapted to the sequence from Vincenzo Librandi, Dec 10 2013

A214367 Denominators of continued fraction convergents to ( sqrt(4*sqrt(3)-3) - 1 )/4.

Original entry on oeis.org

1, 4, 53, 57, 110, 167, 277, 444, 721, 119409, 120130, 239539, 1557364, 1796903, 3354267, 15213971, 109852064, 2871367635, 2981219699, 14796246431, 17777466130, 50351178691, 68128644821, 186608468333, 254737113154, 696082694641, 950819807795, 2597722310231, 13939431358950
Offset: 1

Views

Author

Jean-Claude Babois, Feb 20 2013

Keywords

Comments

This constant arises as the edge-length of one of the pieces in the classical 4-piece dissection of a square to an equilateral triangle (see references in A110312).

Crossrefs

Cf. A224224, A110312.

Programs

  • Mathematica
    Denominator[Convergents[(Sqrt[4 Sqrt[3] - 3] - 1)/4, 50]]

Extensions

Edited by N. J. A. Sloane, Apr 08 2013 at the request of Jean-Claude Babois
Mathematica code corrected to agree with terms by Ray Chandler, Mar 13 2017

A218155 Numbers congruent to 2, 3, 6, 11 mod 12.

Original entry on oeis.org

2, 3, 6, 11, 14, 15, 18, 23, 26, 27, 30, 35, 38, 39, 42, 47, 50, 51, 54, 59, 62, 63, 66, 71, 74, 75, 78, 83, 86, 87, 90, 95, 98, 99, 102, 107, 110, 111, 114, 119, 122, 123, 126, 131, 134, 135, 138, 143, 146, 147, 150, 155, 158, 159, 162, 167, 170, 171, 174
Offset: 1

Views

Author

Jean-Claude Babois, Oct 22 2012

Keywords

Links

Crossrefs

Cf. A042944, A042945.

Programs

  • Mathematica
    LinearRecurrence[{2, -2, 2, -1}, {2, 3, 6, 11}, 100] (* T. D. Noe, Nov 11 2012 *)
  • PARI
    for(m=2,175,if(binomial(m,4)%binomial(m,2)==0,print1(m,", "))) \\ Hugo Pfoertner, Aug 11 2020

Formula

a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) - a(n-4). - Charles R Greathouse IV, Nov 09 2012
G.f.: x^2*(x^3+4*x^2-x+2) / ((x-1)^2*(x^2+1)). - Colin Barker, Jan 07 2013
{m>1|C(m,4)==0 (mod C(m,2))}. - Gary Detlefs, Jan 11 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(3)+1)*Pi/24 - log(2+sqrt(3))/(4*sqrt(3)) - log(2)/6. - Amiram Eldar, Mar 18 2022

Extensions

Edited by Andrey Zabolotskiy, Aug 11 2020

A169888 Number of n-member subsets of 1..2n whose elements sum to a multiple of n.

Original entry on oeis.org

1, 2, 2, 8, 18, 52, 152, 492, 1618, 5408, 18452, 64132, 225432, 800048, 2865228, 10341208, 37568338, 137270956, 504171584, 1860277044, 6892335668, 25631327688, 95640829924, 357975249028, 1343650267288, 5056424257552, 19073789328752, 72108867620204
Offset: 0

Views

Author

N. J. A. Sloane, Jul 07 2010, based on a letter from Jean-Claude Babois

Keywords

Comments

This is twice A145855 (for n>0), which is the main entry for this problem.

Links

Crossrefs

Cf. A061865, A119358, A145855, A318431, A318432, A318433.
Column k=2 of A304482.

Programs

  • Maple
    with(combinat): t0:=[]; for n from 1 to 8 do ans:=0; t1:=choose(2*n,n); for i in t1 do s1:=add(i[j],j=1..n); if s1 mod n = 0 then ans:=ans+1; fi; od: t0:=[op(t0),ans]; od:
  • Mathematica
    a[n_] := Sum[(-1)^(n+d)*EulerPhi[n/d]*Binomial[2d, d]/n, {d, Divisors[n]}]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 22 2012, after T. D. Noe's program in A145855 *)
  • PARI
    a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(n+d)*eulerphi(n/d)*binomial(2*d, d)/n)); \\ Altug Alkan, Aug 27 2018, after T. D. Noe at A145855

Formula

a(n) = A061865(2n,n). - Alois P. Heinz, Aug 28 2018
a(n) ~ 2^(2*n) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 28 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 26 2018

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

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