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 1-4 of 4 results.

A261130 a(n) = Product(p prime | n < p <= 2*n).

Original entry on oeis.org

1, 2, 3, 5, 35, 7, 77, 143, 143, 2431, 46189, 4199, 96577, 7429, 7429, 215441, 6678671, 392863, 392863, 765049, 765049, 31367009, 1348781387, 58642669, 2756205443, 2756205443, 2756205443, 146078888479, 146078888479, 5037203051, 297194980009, 584803025179
Offset: 0

Views

Author

Peter Luschny, Oct 31 2015

Keywords

Comments

Essentially the same as A068111. - R. J. Mathar, Nov 23 2015
a(n) is a divisor of binomial(2*n, n); the quotient binomial(2*n, n) / a(n) is A263931(n). - Robert FERREOL, Sep 03 2022

Examples

			a(0) = 1 because the empty product is 1 by convention.
a(4) = 35 because {p prime | 4 < p <= 8} = {5, 7}.

Crossrefs

Cf. A000984 (binomial(2*n,n)), A034386, A263931, A356637.

Programs

  • Maple
    a := n -> convert(select(isprime, {$n+1..2*n}),`*`):
print(seq(a(n), n=0..31));
  • Mathematica
    Join[{1},Table[Times@@Prime[Range[PrimePi[n]+1,PrimePi[2n]]],{n,40}]] (* Harvey P. Dale, May 09 2017 *)
  • PARI
    A261130(n,P=1)={forprime(p=n+1,2*n,P*=p);P} \\ M. F. Hasler, Nov 25 2015
  • Python
    from sympy import primorial
def A261130(n): return primorial(n<<1,nth=False)//primorial(n,nth=False) if n else 1 # Chai Wah Wu, Sep 07 2022

A068110 Denominators of coefficients in J0(i*sqrt(x))^2 power series where J0 denotes the ordinary Bessel function of order 0.

Original entry on oeis.org

1, 2, 32, 576, 73728, 409600, 176947200, 17340825600, 1183800360960, 1725980926279680, 3451961852559360000, 39779750872350720000, 137478819014844088320000, 1858713633080692074086400, 377800756235068077873561600, 2550155104586709525646540800000, 20890870616774324434096462233600000
Offset: 0

Views

Author

Benoit Cloitre, Mar 21 2002

Keywords

References

  • Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, 1989; see Hypergeometric series, p. 59.

Links

Crossrefs

Cf. A068111 (numerators).

Programs

  • Mathematica
    Denominator[CoefficientList[Series[BesselJ[0, I*Sqrt[x]]^2, {x, 0, 15}], x]] (* Amiram Eldar, Jan 17 2025 *)

Formula

J0(i*sqrt(x))^2 = Sum_{n>=0} (2n)!/(n!)^4/2^(2n)*x^n.

Extensions

More terms from Amiram Eldar, Jan 17 2025

A094337 a(n) = floor((product of composites among next n numbers)/(product of primes among next n numbers)).

Original entry on oeis.org

0, 1, 4, 1, 617, 112, 845, 25376, 2985, 314, 1597052, 138874, 1173486218, 63368255819, 4370224539, 281949970, 5377913733006, 376453961310474, 7345939461247630, 572983277977315172, 27950403803771471, 1300018781570766
Offset: 1

Views

Author

Amarnath Murthy, May 17 2004

Keywords

Comments

The products of primes in the denominators are in A068111 for n>=2. - R. J. Mathar, Jul 27 2007

Examples

			a(3) = floor((4*6)/5) = 4.
a(4) = floor(6*8/(5*7)) = floor(48/35) = 1.
a(5) = floor(6*8*9*10/7) = floor(4320/7) = 617.

Programs

  • Maple
    A094337 := proc(n) local nup,ndown,i ; nup := 0 ; ndown := 0 ; for i from n+1 to 2*n do if isprime(i) then if ndown = 0 then ndown :=i ; else ndown := ndown*i ; fi ; else if nup = 0 then nup := i ; else nup := nup*i ; fi ; fi ; od; floor(nup/ndown) ; end: seq(A094337(n),n=1..24) ; # R. J. Mathar, Jul 27 2007

Extensions

Corrected and extended by R. J. Mathar, Jul 27 2007

A201146 Triangle read by rows: T(n,k) = (n#)/(k#), 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 6, 3, 1, 1, 30, 15, 5, 5, 1, 30, 15, 5, 5, 1, 1, 210, 105, 35, 35, 7, 7, 1, 210, 105, 35, 35, 7, 7, 1, 1, 210, 105, 35, 35, 7, 7, 1, 1, 1, 210, 105, 35, 35, 7, 7, 1, 1, 1, 1, 2310, 1155, 385, 385, 77, 77, 11, 11, 11, 11, 1, 2310, 1155, 385, 385
Offset: 1

Views

Author

Arkadiusz Wesolowski, Nov 27 2011

Keywords

Comments

Row sums give A201156.
Central terms give A068111: T(2*n-1,n) = A068111(n).
T(n,1) = A034386(n).
T(n,n-1) = A089026(n) for n > 1.
T(n,n) = A000012(n).
Let n > 1 and p = A000040(n). Then T(p,p-1) = T(p+1,p-1) = p.
T(2*n-1,n-1) = A073838(n) for n > 1.
T(2*n,n+1) = A144186(n).

Examples

			1:                                   1
2:                               2       1
3:                           6       3       1
4:                       6       3       1       1
5:                   30      15      5       5       1
6:               30      15      5       5       1       1
7:           210     105     35      35      7       7       1
8:       210     105     35      35      7       7       1       1
9:   210     105     35      35      7       7       1       1       1

Links

Crossrefs

Cf. A034386.

Programs

  • Mathematica
    lst = {}; Do[AppendTo[lst, Product[Prime[i], {i, PrimePi[n]}]/Product[Prime[i], {i, PrimePi[k]}]], {n, 12}, {k, n}]; lst (* Arkadiusz Wesolowski, Dec 02 2011 *)
Showing 1-4 of 4 results.

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