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.

A160144 Numerator of (2*n+1)/(2^(2*n+1)-1).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 3, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 9, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 15, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127
Offset: 0

Views

Author

Peter Luschny, May 03 2009

Keywords

Comments

This first differs from A005408 (the odd numbers 2n+1) at a(10). The sequence of differences is A160145. This explains the similarity of A009843 (expansion of x/cos(x)) and A160143. A156769 describes a similar companion to A036279 (expansion of tan(x)).

Links

Crossrefs

Cf. A005408, A009843, A036279, A156769, A160143, A160145, A303449 (denominators).

Programs

  • Magma
    [Numerator((2*n+1)/(2^(2*n+1)-1)): n in [0..70]]; // Vincenzo Librandi, Apr 25 2018
  • Maple
    seq(numer((2*n+1)/(4^(2*n+1)-2^(2*n+1))),n=0..32);
seq(numer((2*n+1)/(2^(2*n+1)-1)),n=0..50); # Altug Alkan, Apr 21 2018
  • Mathematica
    Array[Numerator[(2 # + 1)/(2^(2 # + 1) - 1)] &, 64, 0] (* Michael De Vlieger, Apr 21 2018 *)
  • PARI
    vector(80,n, n--; numerator((2*n+1)/(4^(2*n+1)-2^(2*n+1)))) \\ Michel Marcus, Jan 31 2015
  • PARI
    forstep(k=1, 1e3, 2, print1(numerator(k/(2^k-1)), ", ")); \\ Altug Alkan, Apr 21 2018

Extensions

More terms from Michel Marcus, Jan 31 2015
Name simplified by Altug Alkan, Apr 21 2018
Further edited by N. J. A. Sloane, Apr 24 2018

A160143 a(n) = Numerator((-1)^n*Euler(2*n)*(2*n+1)/(4^(2*n+1)-2^(2*n+1))), where Euler(n) = A122045(n).

Original entry on oeis.org

1, 3, 25, 427, 12465, 555731, 35135945, 2990414715, 329655706465, 45692713833379, 1111113564712575, 1595024111042171723, 387863354088927172625, 110350957750914345093747
Offset: 0

Views

Author

Peter Luschny, May 03 2009

Keywords

Comments

Resembles the coefficients of the series for x/cos(x).
The first difference with sequence A009843 (expansion of x/cos(x)) occurs at a(10). An explanation can be found in the similarity of the numerators of (2*n+1)/(2^(2*n+1)-1) and the odd numbers 2n+1 (cf. A160144).
Similarly, A156769 resembles A036279 (from the expansion of tan(x)).

Crossrefs

Cf. A009843, A122045, A160144, A160145, A036279, A156769.

Programs

  • Maple
    a := n -> (-1)^iquo(n,2)*euler(n)*(n+1)/(4^(n+1)-2^(n+1));
seq(numer(a(2*n)),n=0..13);

A303449 Denominator of (2*n+1)/(2^(2*n+1)-1).

Original entry on oeis.org

1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 299593, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311, 2251799813685247
Offset: 0

Views

Author

Altug Alkan, Apr 24 2018

Keywords

Comments

If A160145(n) = 0, then a(n) = A083420(n).
Least values of k such that a(k) = A083420(k)/A036259(n) are 0, 10, 126, 77, 540, 73, 1242, 328, 1540, 489 for 1 <= n <= 10.

Crossrefs

Cf. A005408, A036259, A083420, A160144 (numerators), A160145.

Programs

  • Maple
    seq(denom((2*n+1)/(2^(2*n+1)-1)), n=0..25);
  • PARI
    a(n) = denominator((2*n+1)/(2^(2*n+1)-1));
  • PARI
    forstep(k=1, 1e2, 2, print1(denominator(k/(2^k-1)), ", "));
Showing 1-3 of 3 results.

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

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