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.

A134652 Indices for which A097344 differs from A097345.

Original entry on oeis.org

59, 1519, 7814, 17225, 39079, 950619, 977019, 1280699
Offset: 1

Views

Author

M. F. Hasler, Jan 25 2008

Keywords

Comments

The terms 59 and 1519 were found by Daniel Glasscock (glasscock(AT)rice.edu), Jan 04 2008.
a(6) > 10^5.
These are the numbers m such that f(m) = Sum_{k=0..m} binomial(m,k)/(k+1)^2 (binomial transform of 1/(k+1)^2) has the same numerator as g(m) = Sum_{k=0..m} (2^(k+1) - 1)/(k+1) (which are also the partial sums of the binomial transformation of 1/(k+1)).
Obviously, f(m) = Sum_{k=0..m} binomial(m+1, k+1)/((k+1)*(m+1)) and since g(m) = (m+1) f(m) (cf. notes by R. J. Mathar on A097345), g(m) = Sum_{k=1..m+1} binomial(m+1,k)/k.
We have the equivalences: numerator(g(n)) = numerator(f(n)) <=> (n+1) | denominator(f(n)) <=> gcd(numerator(g(n)), n+1) = 1.
Therefore this sequence can be alternatively defined in either of the following two ways: numbers n such that the denominator of f(n) is not divisible by (n+1); numbers n such that the numerator of g(n) is not coprime to (n+1).
In terms of M = m+1, the characterization reads: a(n)+1 = numbers M such that denominator(Sum_{k=1..M} binomial(M-1, k-1)/k^2) is not a multiple of M = numbers M such that numerator(Sum_{k=1..M} (2^k - 1)/k) is not coprime to M.

Crossrefs

Programs

  • Mathematica
    Reap[ For[n = 1, n < 10^5, n++, If[ !Divisible[ Denominator[ HypergeometricPFQ[{1, 1, -n}, {2, 2}, -1]], n+1], Print[n]; Sow[n] ] ] ][[2, 1]] (* Jean-François Alcover, Oct 15 2013 *)
  • PARI
    t=1; for( n=2,10^5, gcd( numerator(t+=(1<1 & print(n-1))

Extensions

a(6)-a(8) from Amiram Eldar, Apr 08 2019

A081528 a(n) = n*lcm{1,2,...,n}.

Original entry on oeis.org

1, 4, 18, 48, 300, 360, 2940, 6720, 22680, 25200, 304920, 332640, 4684680, 5045040, 5405400, 11531520, 208288080, 220540320, 4423058640, 4655851200, 4888643760, 5121436320, 123147264240, 128501493120, 669278610000, 696049754400
Offset: 1

Views

Author

Amarnath Murthy, Mar 27 2003

Keywords

Comments

Denominators in binomial transform of 1/(n + 1)^2. - Paul Barry, Aug 06 2004
Construct a sequence S_n from n sequences b_1, b_2, ..., b_n of periods 1, 2, ..., n, respectively, say, b_1 = [1, 1, ...], b_2 = [1, 2, 1, 2, ...], ..., b_n = [1, 2, 3, ..., n, 1, 2, 3, ..., n, ...], by taking S_n = [b_1(1), b_2(1), ..., b_n(1), b_1(2), b_2(2), ..., b_n(2), ..., b_1(n), b_2(n), ..., b_n(n), ...] (by listing the b_i sequences in rows and taking each column in turn as the next n terms of S_n). Then a(n) is the period of sequence S_n. - Rick L. Shepherd, Aug 21 2006
This is a sequence that goes in strictly ascending order. The related sequence A003418 also goes in ascending order but has consecutive repeated terms. Since n increases, then so too does a(n) even when A003418(n) doesn't. - Alonso del Arte, Nov 25 2012

Examples

			a(2) = 4 because the least common multiple of 1 and 2 is 2, and 2 * 2 = 4.
a(3) = 18 because lcm(1,2,3) = 6, and 3 * 6 = 18.
a(4) = 48 because lcm(1, 2, 3, 4) = 12, and 4 * 12 = 48.
		

Crossrefs

Programs

  • Derive
    a(n) := (n + 1)*LCM(VECTOR(k + 1, k, 0, n)) " Paul Barry, Aug 06 2004 "
    
  • Mathematica
    Table[n*LCM@@Range[n], {n, 30}] (* Harvey P. Dale, Oct 09 2012 *)
  • PARI
    l=vector(35); l[1]=1; print1("1, "); for(n=2,35, l[n]=lcm(l[n-1],n); print1(n*l[n],", ")) \\ Rick L. Shepherd, Aug 21 2006

Formula

a(n) = A003418(n) * n. - Martin Fuller, Jan 03 2006

Extensions

More terms from Paul Barry, Aug 06 2004
Entry revised by N. J. A. Sloane, Jan 15 2006

A097345 Numerators of the partial sums of the binomial transform of 1/(n+1).

Original entry on oeis.org

1, 5, 29, 103, 887, 1517, 18239, 63253, 332839, 118127, 2331085, 4222975, 100309579, 184649263, 1710440723, 6372905521, 202804884977, 381240382217, 13667257415003, 25872280345103, 49119954154463, 93501887462903
Offset: 0

Views

Author

Paul Barry, Aug 06 2004

Keywords

Comments

Numerators in the expansion of log((1-x)/(1-2x)) / (1-x) are 0,1,5,29,.. - Paul Barry, Feb 09 2005
Is this identical to A097344? - Aaron Gulliver, Jul 19 2007. The answer turns out to be No - see A134652.
From n=9 on, the putative formula a(n)=A003418(n+1)*sum{k=0..n, (2^(k+1)-1)/(k+1)} is false. The least n for which a(n) is different from A097344(n) is n=59, then they agree again until n=1519. - M. F. Hasler, Jan 25 2008

Crossrefs

Programs

  • Mathematica
    Table[ Sum[(2^(k+1)-1)/(k+1), {k, 0, n}] // Numerator, {n, 0, 21}] (* Jean-François Alcover, Oct 14 2013, after Pari *)
  • PARI
    A097345(n) = numerator(sum(k=0,n,(2^(k+1)-1)/(k+1)))

Extensions

Edited and corrected by Daniel Glasscock (glasscock(AT)rice.edu), Jan 04 2008 and M. F. Hasler, Jan 25 2008
Moved comment concerning numerators of the logarithm from A097344 to here where it is correct - R. J. Mathar, Mar 04 2010
Showing 1-3 of 3 results.