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 1-6 of 6 results.

A014973 a(n) = n / gcd(n, (n-1)!).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1
Offset: 1

Views

Author

Marc LeBrun

Keywords

Comments

Denominator in n!/n^2. Also denominator in Taylor series expansion of dilog function (also called Li_2). - Ralf Stephan, Mar 28 2004

Links

Crossrefs

Cf. A092043.

Programs

  • Magma
    [Denominator(Factorial(n)/n^2): n in [1..80]]; // Vincenzo Librandi, Apr 15 2014
  • Maple
    seq(n / igcd(n, (n-1)!), n = 1..88);  # Peter Luschny, Nov 02 2022
  • Mathematica
    Table[n/GCD[n,(n-1)!],{n,90}] (* Harvey P. Dale, Mar 16 2012 *)
Table[Denominator[n!/n^2], {n, 1, 100}] (* Vincenzo Librandi, Apr 15 2014 *)
  • PARI
    a(n)=numerator(polcoeff((x+1)*exp(x+x*O(x^(n-1))), n-1)); \\ Gerry Martens, Aug 12 2015
  • PARI
    a(n) = { my(f = factor(n), res = n); for(i = 1, #f~, res /= f[i, 1]^(min(f[i, 2], val(n-1, f[i, 1]))) ); res }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Oct 27 2022
  • PARI
    a(n) = if(n == 4, return(2), return(n^isprime(n))) \\ David A. Corneth, Oct 27 2022

Formula

a(4) = 2; otherwise a(n) = 1 unless n is a prime in which case a(n) = n. - Ola Veshta (olaveshta(AT)my-deja.com), May 30 2001
a(n) = denominator((n-1)! * Sum_{i=1..n} (1 - 1/i)). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 16 2004
a(n+1) equals the numerator of the coefficient of x^n in the expansion of (1 + x)*exp(x), with denominator A092043(n+1), for n >= 0. - Wolfdieter Lang, Oct 26 2022
a(n) = denominator((-1)^n*n!/(1+n)). - Stefano Spezia, Jun 24 2024

A170826 a(n) = gcd(n^2, n!).

Original entry on oeis.org

1, 2, 3, 8, 5, 36, 7, 64, 81, 100, 11, 144, 13, 196, 225, 256, 17, 324, 19, 400, 441, 484, 23, 576, 625, 676, 729, 784, 29, 900, 31, 1024, 1089, 1156, 1225, 1296, 37, 1444, 1521, 1600, 41, 1764, 43, 1936, 2025, 2116, 47, 2304, 2401, 2500, 2601, 2704, 53, 2916
Offset: 1

Views

Author

N. J. A. Sloane, Dec 27 2009

Keywords

Crossrefs

Cf. A010051, A092043.

Programs

  • Maple
    GCDWITHFACTORIAL:=proc(a) local b,i,k:
if whattype(a) <> list then RETURN([]); fi:
b:=[]:
for i to nops(a) do b:=[op(b), gcd(a[i],i!)]: od;
RETURN(b);
end:
A170826 := proc(n): gcd(n^2, n!) end: seq(A170826(n), n=1..54); # Johannes W. Meijer, Jun 04 2016
  • Mathematica
    Table[GCD[n^2, n!], {n, 54}] (* Michael De Vlieger, Jun 05 2016 *)
  • PARI
    a(n)=if(isprime(n),n,if(n==4,8,n^2)) \\ Charles R Greathouse IV, Feb 01 2013

Formula

If n is prime then a(n) = n; otherwise, if n <> 4 then a(n) = n^2. - Zak Seidov, Dec 28 2009
a(n) = n!/A092043(n). - Johannes W. Meijer, Jun 04 2016
a(n) = n^2 / n^c(n), where c = A010051 for n >= 5. - Wesley Ivan Hurt, Nov 10 2023

A240533 a(n) = numerators of n!/10^n.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 9, 63, 63, 567, 567, 6237, 18711, 243243, 1702701, 5108103, 5108103, 86837751, 781539759, 14849255421, 14849255421, 311834363841, 3430178002251, 78894094051773, 236682282155319, 236682282155319, 3076869668019147, 83075481036516969
Offset: 0

Views

Author

Vincenzo Librandi, Apr 14 2014

Keywords

Links

Crossrefs

Cf. A000142, A011557, A014973, A074143, A092043, A208529.

Programs

  • Magma
    [Numerator(Factorial(n)/10^n): n in [0..30]];
  • Mathematica
    Table[Numerator[n!/10^n], {n, 0, 30}]

A240534 a(n) = denominators of n!/10^n.

Original entry on oeis.org

1, 10, 50, 500, 1250, 2500, 12500, 125000, 156250, 1562500, 1562500, 15625000, 39062500, 390625000, 1953125000, 3906250000, 2441406250, 24414062500, 122070312500, 1220703125000, 610351562500, 6103515625000, 30517578125000
Offset: 0

Views

Author

Vincenzo Librandi, Apr 14 2014

Keywords

Links

Crossrefs

Cf. A000142, A011557, A014973, A074143, A092043, A208529.

Programs

  • Magma
    [Denominator(Factorial(n)/10^n): n in [0..30]];
  • Mathematica
    Table[Denominator[n!/10^n], {n, 0, 30}]

A371831 a(n) = numerator(Sum_{k=1..n} k^2/k!).

Original entry on oeis.org

0, 1, 3, 9, 31, 43, 217, 3913, 9133, 73067, 1972819, 6576067, 24112247, 372017527, 1612075951, 157983443203, 7109254944151, 37916026368811, 644572448269793, 34806912206568841, 2422459091299663, 7775794614048301, 277759159408419360043, 2036900502328408640323, 46848711553553398727437
Offset: 0

Views

Author

Stefano Spezia, Apr 07 2024

Keywords

Links

Crossrefs

Cf. A019762, A030297, A371832.
Cf. A014973, A092043.

Programs

  • Mathematica
    a[n_]:=Numerator[(2(E*Gamma[n+1,1]-1)-n)/n!]; Array[a,25,0]
  • PARI
    a(n) = numerator(sum(k=1, n, k^2/k!)); \\ Michel Marcus, Apr 07 2024

Formula

a(n) = numerator((2*(e*Gamma(n+1, 1) - 1) - n)/n!).
a(n) = numerator(A030297(n)/n!).
Limit_{n->oo} a(n)/A371832(n) = 2*e = A019762.

A371832 a(n) = denominator(Sum_{k=1..n} k^2/k!).

Original entry on oeis.org

1, 1, 1, 2, 6, 8, 40, 720, 1680, 13440, 362880, 1209600, 4435200, 68428800, 296524800, 29059430400, 1307674368000, 6974263296000, 118562476032000, 6402373705728000, 445586448384000, 1430277488640000, 51090942171709440000, 374666909259202560000, 8617338912961658880000
Offset: 0

Views

Author

Stefano Spezia, Apr 07 2024

Keywords

Links

Crossrefs

Cf. A019762, A030297, A371831.
Cf. A014973, A092043.

Programs

  • Mathematica
    a[n_]:=Denominator[(2(E*Gamma[n+1,1]-1)-n)/n!]; Array[a,25,0]
  • PARI
    a(n) = denominator(sum(k=1, n, k^2/k!)); \\ Michel Marcus, Apr 07 2024

Formula

a(n) = denominator((2*(e*Gamma(n+1, 1) - 1) - n)/n!).
a(n) = denominator(A030297(n)/n!).
Limit_{n->oo} A371831(n)/a(n) = 2*e = A019762.
Showing 1-6 of 6 results.

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

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