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 1-5 of 5 results.

A286178 Denominator of Sum_{i=1..n}(A285388(i)*A285389(i+1))/(A285388(i+1)*A285389(i)).

Original entry on oeis.org

35, 255255, 100280245065, 2152114239059719935, 1952792905443446363385953865, 40347439369046572433179287578305731225, 772786810821609466400679930812513688804332910188025, 73222791895598040395939625423986137213129917738912050041051075
Offset: 1

Views

Author

Ralf Steiner, May 04 2017

Keywords

Comments

Conjecture: the factorization of a(n) contains all primes 5 < p < 2*(n+1)^2.

Crossrefs

Cf. A285388, A285389, A286179 (numerators).

Programs

  • Mathematica
    a388[i_] := Numerator[2^(1 - 2 i^2) i Binomial[2 i^2, i^2]]; a389[i_] := Denominator[2^(1 - 2 i^2) i Binomial[2 i^2, i^2]]; Denominator[Table[Sum[(a388[i]/a389[i])/((a388[i + 1]/a389[i + 1])), {i, 1, n}], {n, 1, 10}]]

A286179 Numerator of Sum_{i=1..n}(A285388(i)*A285389(i+1))/(A285388(i+1)*A285389(i)).

Original entry on oeis.org

32, 484256, 289919598688, 8368036149759509152, 9542829935669464786892890208, 237478537202498785375436854135610527328, 5320823767933620492346565093167366807147013946077792, 577349384176263735966013123947670534373854750755384636719202336
Offset: 1

Views

Author

Ralf Steiner, May 04 2017

Keywords

Comments

Conjecture: floor(a(n)/A286178(n)) = n.

Crossrefs

Cf. A285388, A285389, A286178 (denominators).

Programs

  • Mathematica
    a388[i_] := Numerator[2^(1 - 2 i^2) i Binomial[2 i^2, i^2]]; a389[i_] := Denominator[2^(1 - 2 i^2) i Binomial[2 i^2, i^2]]; Numerator[Table[Sum[(a388[i]/a389[i])/((a388[i + 1]/a389[i + 1])), {i, 1, n}], {n, 1, 10}]]

A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).

Original entry on oeis.org

1, 35, 36465, 300540195, 79006629023595, 331884405207627584403, 22292910726608249789889125025, 11975573020964041433067793888190275875, 411646257111422564507234009694940786177843149765, 56592821660064550728377610673427602421565368547133335525825
Offset: 1

Views

Author

Ralf Steiner, Apr 18 2017

Keywords

Comments

Editorial comment: This sequence arose from Ralf Steiner's attempt to prove Legendre's conjecture that there is a prime between N^2 and (N+1)^2 for all N. - N. J. A. Sloane, May 01 2017

Links

Crossrefs

Cf. A000079, A000265, A056220, A060757, A201555, A285389 (denominators), A285406, A280655 (similar), A190732 (2/sqrt(Pi)), A285738 (greatest prime factor), A285717, A285730, A285786, A286264, A000290 (n^2), A056220 (2*n^2 -1), A286127 (sum a(n-1)/a(n)).

Programs

  • Magma
    [Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021
  • Mathematica
    Table[Numerator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n],{n,1,10}]
Numerator[Table[2^(1-2 n^2) n Binomial[2 n^2,n^2],{n,1,10}]] (* Ralf Steiner, Apr 22 2017 *)
  • PARI
    A285388(n) = numerator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)); \\ Antti Karttunen, Apr 27 2017
  • PARI
    a(n) = m=n*binomial(2*n^2, n^2);m>>valuation(m,2) \\ David A. Corneth, Apr 27 2017
  • Python
    from sympy import binomial, Integer
def a(n): return (Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).numerator # Indranil Ghosh, Apr 27 2017
  • Sage
    [numerator( n*(n^2+1)*catalan_number(n^2)/2^(2*n^2-1) ) for n in (1..20)] # G. C. Greubel, Dec 11 2021

Formula

a(n) is numerator of n*binomial(2 n^2, n^2)/2^(2*n^2 - 1). - Ralf Steiner, Apr 26 2017
a(n) = numerator(n*A201555(n) / (A060757(n)/2)) = n*A201555(n) / 2^(A285717(n)) = A000265(n*A201555(n)). [Using Ralf Steiner's formula and A285717(n) <= A056220(n), cf. A285406.] - Antti Karttunen, Apr 27 2017
Limit_{i->oo} a(i)*A285389(i+1)/(a(i+1)*A285389(i)) = 1. - Ralf Steiner, May 03 2017

Extensions

Edited (including the removal of the author's claim that this leads to a proof of the Legendre conjecture) by N. J. A. Sloane, May 01 2017
Formula section edited by M. F. Hasler, May 02 2017
Edited by N. J. A. Sloane, May 10 2017

A190732 Decimal expansion of 2/sqrt(Pi).

Original entry on oeis.org

1, 1, 2, 8, 3, 7, 9, 1, 6, 7, 0, 9, 5, 5, 1, 2, 5, 7, 3, 8, 9, 6, 1, 5, 8, 9, 0, 3, 1, 2, 1, 5, 4, 5, 1, 7, 1, 6, 8, 8, 1, 0, 1, 2, 5, 8, 6, 5, 7, 9, 9, 7, 7, 1, 3, 6, 8, 8, 1, 7, 1, 4, 4, 3, 4, 2, 1, 2, 8, 4, 9, 3, 6, 8, 8, 2
Offset: 1

Views

Author

Alonso del Arte, May 17 2011

Keywords

Comments

According to Weisstein, some mathematicians define erf(z) without reference to this constant.
Also equals the average absolute value of the difference of two independent normally distributed random numbers with mean 0 and variance 1. - Jean-François Alcover, Oct 31 2014
Limit_{n->oo} 2^(1-2*n^2)*n*binomial(2*n^2, n^2) is proper to compute this constant (and also Pi) in a base of power 2. - Ralf Steiner, Apr 23 2017
A gauge point marked "c" on slide rule calculating devices in the 20th century. The Pickworth reference notes its use "in calculating the contents of cylinders". - Peter Munn, Aug 14 2020

Examples

			1.12837916709551257...

References

  • Chi Keung Cheung et al., Getting Started with Mathematica, 2nd Ed. New York: J. Wiley (2005) p. 79.
  • C. N. Pickworth, The Slide Rule, 24th Ed., Pitman, London (1945), p 53, Gauge Points.

Links

Crossrefs

Cf. A002161, A019704, A285388, A285389.

Programs

  • Mathematica
    RealDigits[2/Sqrt[Pi], 10, 100][[1]]
RealDigits[Limit[2^(1 - 2 m^2) m Binomial[2 m^2, m^2], m -> Infinity], 10, 100][[1]] (* Ralf Steiner, Apr 22 2017 *)
  • PARI
    2/sqrt(Pi) \\ G. C. Greubel, Jan 09 2017

Formula

Equals Sum_{n>=0} (-1)^n*Gamma((n+1)/2)/Gamma(n/2+1). - Jean-François Alcover, Jun 12 2013
Equals 1/A019704. - Michel Marcus, Jan 09 2017
Equals Limit_{n->infinity} A285388(n)/A285389(n). - Ralf Steiner, Apr 22 2017

A285406 Base-2 logarithm of denominator of Sum_{k=0..n^2-1} (-1)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k)*n).

Original entry on oeis.org

0, 5, 15, 28, 46, 68, 94, 123, 158, 195, 236, 283, 333, 387, 445, 506, 574, 643, 716, 794, 875, 961, 1054, 1146, 1244, 1346, 1451, 1562, 1676, 1794, 1916, 2041, 2174, 2307, 2444, 2586, 2731, 2881, 3034, 3193, 3356, 3520, 3690, 3864, 4041, 4227, 4413, 4601, 4796, 4993
Offset: 1

Views

Author

Ralf Steiner, Apr 18 2017

Keywords

Comments

Needed for studying of Wallis-kind products of central binomials.

Crossrefs

Cf. A000120, A000523, A000984, A007814, A011371, A056220, A201555, A285389, A281264, A285717.

Programs

  • Mathematica
    Log[2,Table[Denominator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n], {n,1,50}]]
Log[2,Denominator[Table[2^(1-2 n^2) n Binomial[2 n^2,n^2],{n,1,50}]]] (* Ralf Steiner, Apr 22 2017 *)
  • PARI
    a(n) = logint(denominator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)), 2); \\ Indranil Ghosh, Apr 27 2017
  • PARI
    val(n, p) = my(r=0); while(n, r+=n\=p);r
a(n) = 2*n^2-1 - valuation(n, 2) - val(2*n^2, 2) + 2*val(n^2, 2) \\ David A. Corneth, Apr 28 2017
  • Python
    from sympy import binomial, integer_log, Integer
def a(n): return integer_log((Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).denominator, 2)[0] # Indranil Ghosh, Apr 27 2017
  • Scheme
    (define (A285406 n) (- (* 2 n n) (A007814 n) (A000120 (* n n)) 1)) ;; Antti Karttunen, Apr 28 2017

Formula

a(n) = A000523(A285389(n)).
a(n) = A056220(n) - A285717(n) = (2*(n^2)) - A007814(n) - A000120(n^2) - 1. - Antti Karttunen, Apr 28 2017, based on Vladimir Shevelev's Jul 20 2009 formula in A000984
Showing 1-5 of 5 results.

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

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