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.

A132050 Denominator of 2*n*A000111(n-1)/A000111(n): approximations of Pi using Euler (up/down) numbers.

Original entry on oeis.org

1, 1, 1, 5, 8, 61, 136, 1385, 3968, 50521, 176896, 2702765, 260096, 199360981, 951878656, 19391512145, 104932671488, 2404879675441, 14544442556416, 74074237647505, 2475749026562048, 69348874393137901, 507711943253426176
Offset: 1

Views

Author

Wolfdieter Lang, Sep 14 2007

Keywords

Comments

The rationals r(n)=2*n*e(n-1)/e(n), where e(n)=A000111(n), approximate Pi as n -> oo. - M. F. Hasler, Apr 03 2013
Numerators are given in A132049.
See the Delahaye reference and a link by W. Lang given in A132049.
From Paul Curtz, Mar 17 2013: (Start)
Apply the Akiyama-Tanigawa transform (or algorithm) to A046978(n+2)/A016116(n+1):
1, 1/2, 0, -1/4, -1/4, -1/8, 0, 1/16, 1/16;
1/2, 1, 3/4, 0, -5/8, -3/4, -7/16, 0; = Balmer0(n)
-1/2, 1/2, 9/4, 5/2, 5/8, -15/8, -49/16;
-1, -7/2, -3/4, 15/2, 25/2, 57/8;
5/2, -11/2, -99/4, -20, 215/8;
8, 77/2, -57/4, -375/2;
-61/2, 211/2, 2079/4;
-136, -1657/2;
1385/2;
The first column is PIEULER(n) = 1, 1/2, -1/2, -1, 5/2, 8, -61/2, -136, 1385/2,... = c(n)/d(n). Abs c(n+1)=1,1,1,5,8,61,... =a(n) with offset=1.
For numerators of Balmer0(n) see A076109, A000265 and A061037(n-1) (End).
Other completely unrelated rational approximations of Pi are given by A063674/A063673 and other references there. - M. F. Hasler, Apr 03 2013

Examples

			Rationals r(n): [2, 4, 3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521, ...].

Crossrefs

Cf. triangle A008281 (main diagonal give zig-zag numbers A000111).

Programs

  • Mathematica
    e[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1])/(n + 1)]]; r[n_] := 2*n*(e[n - 1]/e[n]); a[n_] := Denominator[r[n]]; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Mar 26 2013 *)
  • Python
    from itertools import count, islice, accumulate
from fractions import Fraction
def A132050_gen(): # generator of terms
    yield 1
    blist = (0,1)
    for n in count(2):
        yield Fraction(2*n*blist[-1],(blist:=tuple(accumulate(reversed(blist),initial=0)))[-1]).denominator
A132050_list = list(islice(A132050_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

Formula

a(n)=denominator(r(n)) with the rationals r(n):=2*n*e(n-1)/e(n) where e(n):=A000111(n).

Extensions

Definition made more explicit, and initial terms a(1)=a(2)=1 added by M. F. Hasler, Apr 03 2013

A076107 First of n consecutive integers whose sum is a positive n-th power, or 0 if no such integers exist.

Original entry on oeis.org

1, 0, 8, 0, 623, 119, 117646, 0, 2183, 976558, 25937424596, 0, 23298085122475, 48444505197, 29192926025390618, 0, 48661191875666868473, 21523352, 104127350297911241532832, 0, 278218429446951548637196391
Offset: 1

Views

Author

Amarnath Murthy, Oct 08 2002

Keywords

Comments

No sum exists precisely when n == 0 (mod 4). a(2) = 0 is a legitimate value.
The sum is given by A076108(n) = A076109(n)^n for n != 0 (mod 4).
a(p) = p^(p-1) - (p-1)/2 for prime p.

Examples

			a(3) = 8 as 8+9+10 = 27 = 3^3. a(6) = 119 as 119+120+...+124 = 729 = 3^6.

Crossrefs

Cf. A076108, A076109.

Programs

  • PARI
    for(n=1,30,t=n*(n-1)/2;f=0;for(r=1,10^4,if((r^n-t)%n==0,f=(r^n-t)/n;break));print1(f","))
  • PARI
    {A076107(n)=if(n%4==0,return(0));m=n;if(m%2==0,m\=2);f=factorint(m)[,1];p=1;(2*prod(i=1,length(f),f[i])^n/n-n+1)/2} (Alekseyev)

Formula

a(n) = A076108(n)/n - (n-1)/2 for n != 0 (mod 4).
a(4k)=0; otherwise a(n) = (2*A076108(n)/n - n + 1)/2 = (2*p1^n*...*pm^n/n - n + 1)/2 where p1, ..., pm are all distinct odd primes dividing n. - Max Alekseyev, Jun 10 2005

Extensions

Corrected and extended by Ralf Stephan, Mar 30 2003
Revised by Max Alekseyev and David W. Wilson, Jun 10 2005
More terms from Max Alekseyev, Jun 10 2005

A076108 Least positive n-th power that is the sum of n consecutive integers, or 0 if no such n-th power exists.

Original entry on oeis.org

1, 1, 27, 0, 3125, 729, 823543, 0, 19683, 9765625, 285311670611, 0, 302875106592253, 678223072849, 437893890380859375, 0, 827240261886336764177, 387420489, 1978419655660313589123979, 0
Offset: 1

Views

Author

Amarnath Murthy, Oct 08 2002

Keywords

Comments

No n-th power exists precisely when n == 0 (mod 4).
The first term of the sum is A076107(n) for n != 0 (mod 4).
a(p) = p^p for prime p.

Examples

			27 = 3^3 = 8+9+10 is least positive cube that is sum of 3 consecutive integers, hence a(3) = 27.

Crossrefs

Cf. A076107, A076109.

Programs

  • PARI
    for(n=1,30,t=n*(n-1)/2:f=0:for(r=1,10^4,if((r^n-t)%n==0,f=r^n:break)):print1(f","))
  • PARI
    {A076108(n)=if(n%4==0,return(0));m=n;if(m%2==0,m\=2);f=factorint(m)[,1];prod(i=1,length(f),f[i])^n} (Alekseyev)

Formula

a(n) = n*A076107(n)+(n^2-n)/2 for n != 0 (mod 4).
a(n) = A076109(n)^n.
a(4k)=0; otherwise a(n)=p1^n*...*pm^n where p1, ..., pm are all distinct odd primes dividing n. - Max Alekseyev, Jun 10 2005

Extensions

Corrected and extended by Ralf Stephan, Mar 30 2003
Revised by Max Alekseyev and David W. Wilson, Jun 10 2005
More terms from Max Alekseyev, Jun 10 2005
Showing 1-3 of 3 results.

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