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 1-3 of 3 results.

A073076 Numbers k such that 2*k+1 divides C(2*k,k).

Original entry on oeis.org

97, 136, 178, 192, 199, 292, 313, 332, 448, 467, 472, 478, 487, 535, 542, 577, 604, 617, 697, 773, 790, 797, 852, 885, 940, 962, 967, 997, 1017, 1045, 1096, 1127, 1147, 1165, 1182, 1202, 1237, 1291, 1292, 1319, 1332, 1339, 1345, 1354, 1368, 1397, 1414
Offset: 1

Views

Author

Benoit Cloitre, Aug 17 2002

Keywords

Comments

Integers k such that A056617(k) = 1. - Michel Marcus, May 27 2019
Numbers n such that A005408(n) divides A000984(n). - Felix Fröhlich, May 27 2019

Links

Crossrefs

Cf. A000984, A005408, A056616, A056617, A080394.

Programs

  • Mathematica
    Select[Range@ 1500, Mod[Binomial[2 #, #], 2 # + 1] == 0 &] (* Michael De Vlieger, May 27 2019 *)
  • PARI
    isok(n) = ! (binomial(2*n, n) % (2*n+1)); \\ Michel Marcus, Nov 28 2013

A056617 Denominator of binomial(2*n,n) / (2*n+1).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 5, 17, 19, 21, 23, 25, 27, 29, 31, 11, 7, 37, 13, 41, 43, 3, 47, 49, 17, 53, 55, 57, 59, 61, 9, 65, 67, 23, 71, 73, 75, 11, 79, 81, 83, 17, 29, 89, 13, 31, 19, 97, 11, 101, 103, 35, 107, 109, 37, 113, 115, 39, 119, 121, 41, 125
Offset: 0

Views

Author

N. J. A. Sloane, Aug 28 2000

Keywords

Comments

The numerators are given in A056616.

Examples

			The rationals r(n) begin: 1, 2/3, 6/5, 20/7, 70/9, 252/11, 924/13, 1144/5, 12870/17, ...

Links

Crossrefs

Cf. A056616, A000108.

Programs

  • Magma
    [Denominator((Binomial (2*n, n)) / (2*n + 1)): n in [0..70]]; // Vincenzo Librandi, May 27 2019
  • Mathematica
    Table[Binomial[2 n, n]/(2 n + 1), {n, 0, 70}]//Denominator (* Harvey P. Dale, May 01 2019 *)
  • PARI
    a(n) = denominator(binomial(2*n,n) / (2*n+1)); \\ Michel Marcus, May 27 2019

Formula

a(n) = denominator(r(n)) with r(n) = binomial(2*n,n)/(2*n+1).
G.f. of r(n): 1/(2*sqrt(x))*arcsin(2*sqrt(x)). [Vladimir Kruchinin, May 31 2013]

A101681 Numbers k such that gcd(C(2k,k), 2k+1) > 1.

Original entry on oeis.org

7, 16, 17, 19, 22, 25, 31, 34, 38, 42, 43, 45, 46, 47, 49, 52, 55, 58, 61, 64, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 87, 88, 92, 93, 94, 97, 100, 102, 103, 104, 106, 107, 110, 112, 115, 117, 122, 123, 124, 127, 129, 130, 133, 136, 139, 142, 143, 145, 147, 148
Offset: 1

Views

Author

Ralf Stephan, Dec 11 2004

Keywords

Comments

Positions where A056616 differs from A000984.
The set seems to have greater cardinality than its complement.
Positions where A055786 differs from A001790. - Mohammed Yaseen, Aug 03 2024

Examples

			7 is in the sequence as gcd(binomial(2*7, 7), 2*7 + 1) = gcd(3432, 15) = gcd(3*1144, 3*5) > 1. - _David A. Corneth_, Apr 03 2021

Links

Crossrefs

Cf. A000984, A056616, A126786.

Programs

  • Mathematica
    Select[Range[200],GCD[Binomial[2 #,#],2 #+1]>1&] (* Harvey P. Dale, May 11 2019 *)
  • PARI
    is(n) = { my(f = factor(2*n+1)); for(i = 1, #f~, if(val(2*n, f[i, 1])-2*val(n, f[i, 1]) > 0, return(1))); 0 }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Apr 03 2021
Showing 1-3 of 3 results.

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

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