A145609 -id:A145609 - cp's OEIS frontend

A145737 a(n) = square part of A145609(n).

1, 5, 7, 1, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1, 163

Offset: 1

Artur Jasinski, Oct 17 2008

nonn

For squarefree parts see A145738. A128059 is a very similar sequence.

Cf. A008833, A128059, A145609, A145738.

seq(denom(n!*3*n*(n+1)/(2*(2*n+1))), n=1..81); # Gary Detlefs, Oct 18 2011

m = 1; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; b = Sqrt[Numerator[k]] /. Sqrt[] -> 1; AppendTo[aa, b], {r, 1, 137}]; aa (* _Artur Jasinski *)

from sympy import isprime
def A145737(n): return a if isprime(a:=(n<<1)+1) and n>1 else 1 # Chai Wah Wu, Feb 26 2024

a(n) = 2n+1 if 2n+1 is prime, 1 otherwise, for n > 1.
From Gary Detlefs, Oct 18 2011: (Start)
a(n) = Denominator(n!*(Sum_{k=1..n} k^3)/(Sum_{k=1..n} k^2))
     = Denominator(n!*3*n*(n+1)/(2*(2*n+1))). (End)

A145738 a(n) = squarefree part of A145609(n).

Original entry on oeis.org

3, 1, 1, 761, 61, 509, 1171733, 8431, 39541, 55835135, 36093, 1347822955, 34395742267, 375035183, 9682292227, 586061125622639, 54062195834749, 40030624861, 2053580969474233, 1236275063173, 6657281227331

Offset: 1

Artur Jasinski, Oct 17 2008

nonn

Cf. A128059, A145609, A145738, A145737 (for square parts).

m = 1; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; b = Sqrt[Numerator[k]] /. Sqrt[] -> 1; c = Sqrt[Numerator[k]]/b /. Sqrt[x] -> x; AppendTo[aa, c], {r, 1, 25}]; aa

A001008 a(n) = numerator of harmonic number H(n) = Sum_{i=1..n} 1/i.

Original entry on oeis.org

1, 3, 11, 25, 137, 49, 363, 761, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 14274301, 275295799, 55835135, 18858053, 19093197, 444316699, 1347822955, 34052522467, 34395742267, 312536252003, 315404588903, 9227046511387

Offset: 1

N. J. A. Sloane

H(n)/2 is the maximal distance that a stack of n cards can project beyond the edge of a table without toppling.
By Wolstenholme's theorem, p^2 divides a(p-1) for all primes p > 3.
From Alexander Adamchuk, Dec 11 2006: (Start)
p divides a(p^2-1) for all primes p > 3.
p divides a((p-1)/2) for primes p in A001220.
p divides a((p+1)/2) or a((p-3)/2) for primes p in A125854.
a(n) is prime for n in A056903. Corresponding primes are given by A067657. (End)
a(n+1) is the numerator of the polynomial A[1, n](1) where the polynomial A[genus 1, level n](m) is defined to be Sum_{d = 1..n - 1} m^(n - d)/d. (See the Mathematica procedure generating A[1, n](m) below.) - Artur Jasinski, Oct 16 2008
Better solutions to the card stacking problem have been found by M. Paterson and U. Zwick (see link). - Hugo Pfoertner, Jan 01 2012
a(n) = A213999(n, n-1). - Reinhard Zumkeller, Jul 03 2012
a(n) coincides with A175441(n) if and only if n is not from the sequence A256102. The quotient a(n) / A175441(n) for n in A256102 is given as corresponding entry of A256103. - Wolfdieter Lang, Apr 23 2015
For a very short proof that the Harmonic series diverges, see the Goldmakher link. - N. J. A. Sloane, Nov 09 2015
All terms are odd while corresponding denominators (A002805) are all even for n > 1 (proof in Pólya and Szegő). - Bernard Schott, Dec 24 2021

			H(n) = [ 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, ... ].
Coincidences with A175441: the first 19 entries coincide because 20 is the first entry of A256102. Indeed, a(20)/A175441(20) = 55835135 / 11167027 = 5 = A256103(1). - _Wolfdieter Lang_, Apr 23 2015

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 258-261.
H. W. Gould, Combinatorial Identities, Morgantown Printing and Binding Co., 1972, # 1.45, page 6, #3.122, page 36.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 347.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 615.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, volume II, Springer, reprint of the 1976 edition, 1998, problem 251, p. 154.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Kenny Lau, Table of n, a(n) for n = 1..2295 (first 200 terms provided by T. D. Noe)
David H. Bailey, Jonathan M. Borwein, and Roland Girgensohn, Experimental evaluation of Euler sums, Exper. Math. 3(1) (1994), 17-30; they evaluate the constants Sum_{k>=1} H_k^m/(k+1)^n.
Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.
Hongwei Chen, Interesting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers, J. Int. Seq. 19 (2016), #16.1.5.
R. M. Dickau, Harmonic numbers and the book-stacking problem.
Leo Goldmakher, A short(er) proof of the divergence of the harmonic series.
Antal Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013), 54-82.
Fredrik Johansson, How (not) to compute harmonic numbers. Feb 21 2009.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Hisanori Mishima, Factorizations of Wolstenholme numbers, n=1..100, n=101..200, n=201..300.
Mike Paterson et al., Maximum Overhang.
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.
Peter Shiu, The denominators of harmonic numbers, arXiv:1607.02863 [math.NT], 2016.
N. J. A. Sloane, Illustration of initial terms.
Jonathan Sondow and Eric W. Weisstein, MathWorld: Harmonic Number.
Roberto Tauraso, Some Congruences for Central Binomial Sums Involving Fibonacci and Lucas Numbers, JIS 19 (2016) # 16.5.4.
Eric Weisstein's World of Mathematics, Book Stacking Problem, Wolstenholme's Theorem, Harmonic Mean, Digamma Function.
Wikipedia, Harmonic number.

Cf. A002805 (denominators), A007406, A007408, A007410, A075135, A001220, A125854, A121999, A014566, A056903, A067657, A177427, A177690.
Cf. A145609-A145640. - Artur Jasinski, Oct 16 2008
Cf. A003506. - Paul Curtz, Nov 30 2013
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.
Cf. A195505.

List([1..30],n->NumeratorRat(Sum([1..n],i->1/i))); # Muniru A Asiru, Dec 20 2018

import Data.Ratio ((%), numerator)
a001008 = numerator . sum . map (1 %) . enumFromTo 1
a001008_list = map numerator $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012

[Numerator(HarmonicNumber(n)): n in [1..30]]; // Bruno Berselli, Feb 17 2016

A001008 := proc(n)
    add(1/k,k=1..n) ;
    numer(%) ;
end proc:
seq( A001008(n),n=1..40) ; # Zerinvary Lajos, Mar 28 2007; R. J. Mathar, Dec 02 2016

Table[Numerator[HarmonicNumber[n]], {n, 30}]
(* Procedure generating A[1,n](m) (see Comments section) *) m =1; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, k], {r, 1, 20}]; aa (* Artur Jasinski, Oct 16 2008 *)
Numerator[Accumulate[1/Range[25]]] (* Alonso del Arte, Nov 21 2018 *)
Numerator[Table[((n - 1)/2)*HypergeometricPFQ[{1, 1, 2 - n}, {2, 3}, 1] + 1, {n, 1, 29}]] (* Artur Jasinski, Jan 08 2021 *)

A001008(n) = numerator(sum(i=1,n,1/i)) \\ Michael B. Porter, Dec 08 2009

H1008=List(1); A001008(n)={for(k=#H1008,n-1,listput(H1008,H1008[k]+1/(k+1))); numerator(H1008[n])} \\ about 100x faster for n=1..1500. - M. F. Hasler, Jul 03 2019

from sympy import Integer
[sum(1/Integer(i) for i in range(1, n + 1)).numerator() for n in range(1, 31)]  # Indranil Ghosh, Mar 23 2017

def harmonic(a, b):  # See the F. Johansson link.
    if b - a == 1:
        return 1, a
    m = (a+b)//2
    p, q = harmonic(a,m)
    r, s = harmonic(m,b)
    return p*s+q*r, q*s
def A001008(n): H = harmonic(1,n+1); return numerator(H[0]/H[1])
[A001008(n) for n in (1..29)] # Peter Luschny, Sep 01 2012

H(n) ~ log n + gamma + O(1/n). [See Hardy and Wright, Th. 422.]
log n + gamma - 1/n < H(n) < log n + gamma + 1/n [follows easily from Hardy and Wright, Th. 422]. - David Applegate and N. J. A. Sloane, Oct 14 2008
G.f. for H(n): log(1-x)/(x-1). - Benoit Cloitre, Jun 15 2003
H(n) = sqrt(Sum_{i = 1..n} Sum_{j = 1..n} 1/(i*j)). - Alexander Adamchuk, Oct 24 2004
a(n) is the numerator of Gamma/n + Psi(1 + n)/n = Gamma + Psi(n), where Psi is the digamma function. - Artur Jasinski, Nov 02 2008
H(n) = 3/2 + 2*Sum_{k = 0..n-3} binomial(k+2, 2)/((n-2-k)*(n-1)*n), n > 1. - Gary Detlefs, Aug 02 2011
H(n) = (-1)^(n-1)*(n+1)*n*Sum_{k = 0..n-1} k!*Stirling2(n-1, k) * Stirling1(n+k+1,n+1)/(n+k+1)!. - Vladimir Kruchinin, Feb 05 2013
H(n) = n*Sum_{k = 0..n-1} (-1)^k*binomial(n-1,k)/(k+1)^2. (Wenchang Chu) - Gary Detlefs, Apr 13 2013
H(n) = (1/2)*Sum_{k = 1..n} (-1)^(k-1)*binomial(n,k)*binomial(n+k, k)/k. (H. W. Gould) - Gary Detlefs, Apr 13 2013
E.g.f. for H(n) = a(n)/A002805(n): (gamma + log(x) - Ei(-x)) * exp(x), where gamma is the Euler-Mascheroni constant, and Ei(x) is the exponential integral. - Vladimir Reshetnikov, Apr 24 2013
H(n) = residue((psi(-s)+gamma)^2/2, {s, n}) where psi is the digamma function and gamma is the Euler-Mascheroni constant. - Jean-François Alcover, Feb 19 2014
H(n) = Sum_{m >= 1} n/(m^2 + n*m) = gamma + digamma(1+n), numerators and denominators. (see Mathworld link on Digamma). - Richard R. Forberg, Jan 18 2015
H(n) = (1/2) Sum_{j >= 1} Sum_{k = 1..n} ((1 - 2*k + 2*n)/((-1 + k + j*n)*(k + j*n))) + log(n) + 1/(2*n). - Dimitri Papadopoulos, Jan 13 2016
H(n) = (n!)^2*Sum_{k = 1..n} 1/(k*(n-k)!*(n+k)!). - Vladimir Kruchinin, Mar 31 2016
a(n) = Stirling1(n+1, 2) / gcd(Stirling1(n+1, 2), n!) = A000254(n) / gcd(A000254(n), n!). - Max Alekseyev, Mar 01 2018
From Peter Bala, Jan 31 2019: (Start)
H(n) = 1 + (1 + 1/2)*(n-1)/(n+1) + (1/2 + 1/3)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3 + 1/4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... .
H(n)/n  = 1 + (1/2^2 - 1)*(n-1)/(n+1) + (1/3^2 - 1/2^2)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/4^2 - 1/3^2)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... .
For odd n >= 3, (1/2)*H((n-1)/2) = (n-1)/(n+1) + (1/2)*(n-1)*(n-3)/((n+1)*(n+3)) + 1/3*(n-1)*(n-3)*(n-5)/((n+1)*(n+3)*(n+5)) + ... . Cf. A195505. See the Bala link in A036970. (End)
H(n) = ((n-1)/2) * hypergeom([1,1,2-n], [2,3], 1) + 1. - Artur Jasinski, Jan 08 2021
Conjecture: for nonzero m, H(n) = (1/m)*Sum_{k = 1..n} ((-1)^(k+1)/k) * binomial(m*k,k)*binomial(n+(m-1)*k,n-k). The case m = 1 is well-known; the case m = 2 is given above by Detlefs (dated Apr 13 2013). - Peter Bala, Mar 04 2022
a(n) = the (reduced) numerator of the continued fraction 1/(1 - 1^2/(3 - 2^2/(5 - 3^2/(7 - ... - (n-1)^2/(2*n-1))))). - Peter Bala, Feb 18 2024
H(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)/k (H. W. Gould). - Gary Detlefs, May 28 2024

Edited by Max Alekseyev, Oct 21 2011

Changed title, deleting the incorrect name "Wolstenholme numbers" which conflicted with the definition of the latter in both Weisstein's World of Mathematics and in Wikipedia, as well as with OEIS A007406. - Stanislav Sykora, Mar 25 2016

A145640 Denominator the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=16.

Original entry on oeis.org

1, 3, 5, 35, 315, 3465, 45045, 45045, 255255, 969969, 29393, 2028117, 50702925, 456326325, 13233463425, 820474732350, 4512611027925, 4512611027925, 166966608033225, 12843585233325, 526586994566325, 22643240766351975

Offset: 1

Artur Jasinski, Oct 14 2008

For numerators see A145639. For general properties of A_l(x) see A145609.

Cf. A145609 - A145640.

m = 16; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Denominator[k]], {r, 1, 30}]; aa (*Artur Jasinski*)

Edited by R. J. Mathar, Aug 21 2009

A145656 a(n) = numerator of polynomial of genus 1 and level n for m = 2.

Original entry on oeis.org

0, 2, 5, 32, 131, 661, 1327, 18608, 148969, 447047, 89422, 1967410, 7869871, 102309709, 204620705, 2046213056, 32739453941, 556571077357, 556571247527, 10574855234543, 42299423848079, 42299425233749, 84598851790183

Offset: 1

Artur Jasinski, Oct 16 2008

For the numerators of the polynomials of genus 1 and level n for m = 1 see A001008.
Definition: The polynomial A[1,2*n+1](m) = A[genus 1,level n] is here defined as Sum_{d = 1..n-1} m^(n-d)/d.
First few A[1,n](m):
  n = 1: A[1,1](m) = 0
  n = 2: A[1,2](m) = m
  n = 3: A[1,3](m) = m/2 + m^2
  n = 4: A[1,4](m) = m/3 + m^2/2 + m^3
  n = 5: A[1,5](m) = m/4 + m^2/3 + m^3/2 + m^4
The general formula which uses these polynomials is the following:
(1/(n+1))*Hypergeometric2F1[1, n, n+1, 1/m] = Sum_{k >= 0} m^(-k)/(k + n) = m^n * ArcTanh[(2*m-1)/(2*m^2-2*m+1)] - A[1,n](m) = (m^n)*Log[m/(m-1)] - A[1,n](m).
Conjecture: a(n) = numerator( (2^n)*log(2) - 2^(n+1)*Integral_{x = 0..1} x^(2*n-1)/(1 + x^2)^n ). - Peter Bala, Jun 10 2024
a(n) appears to be a multiple of A068566(n): The sequence {a(n)/A068566(n) : n >= 2} begins [2, 1, 16, 1, 1, 1, 16, 1, 1, 2, 2, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, ...]. - Peter Bala, Aug 07 2025

Cf. A145609-A145640.

A145656 := proc(n) add( 2^(n-d)/d, d = 1..n-1) end: seq(numer(A145656(n)), n = 1..20); # R. J. Mathar, Feb 01 2011

m = 2; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa
(* or *)
a[n_]:=2Integrate[(2-x^n)/(2-x),{x,0,1}]+4(2^(n-1)-1)Log[2]
Table[a[n] // Simplify // Numerator,{n,0,22}] (* Gerry Martens, Jun 04 2016 *)

A145660 a(n) = numerator of polynomial of genus 1 and level n for m = 4 = A[1,n](4).

Original entry on oeis.org

0, 4, 18, 220, 883, 17672, 23566, 659868, 5278979, 95021762, 380087174, 16723836916, 66895348819, 3478558152448, 13914232622662, 11131386100532, 178102177617521, 3027737019533893, 4036982692723202, 306810684647167556

Offset: 1

Artur Jasinski, Oct 16 2008

For numerator of polynomial of genus 1 and level n for m = 1 see A001008.
Definition: The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as
Sum_{d,1,n-1} m^(n-d)/d.
Few first A[1,n](m):
n=1: A[1,1](m)= 0;
n=2: A[1,2](m)= m;
n=3: A[1,3](m)= m/2 + m^2;
n=4: A[1,4](m)= m/4 + m^2/3 + m^3/2 + m^4.
General formula which uses these polynomials is following:
(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =
Sum_{x>=0} m^(-x)/(x+n) =
m^n*arctanh((2m-1)/(2m^2-2m+1)) - A[1,n](m) =
m^n*log(m/(m-1)) - A[1,n](m).

Cf. A145609-A145640, A145656, A145668, A145662, A145664, A145666.

A145660 := proc(n) add( 4^(n-d)/d,d=1..n-1) ; numer(%) ; end proc: # R. J. Mathar, Feb 01 2011

m = 4; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa

A145662 a(n) = numerator of polynomial of genus 1 and level n for m = 5 = A[1,n](5).

Original entry on oeis.org

0, 5, 55, 835, 8365, 41837, 209195, 7321885, 73218955, 1098284605, 5491423277, 302028282755, 1510141416085, 98159192073245, 490795960391965, 2453979801983849, 24539798019883535, 2085882831690821195

Offset: 1

Artur Jasinski, Oct 16 2008

For numerator of polynomial of genus 1 and level n for m = 1 see A001008
Definition: The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as
Sum_{d=1..n-1} m^(n-d)/d.
Few first A[1,n](m):
n=1: A[1,1](m)= 0;
n=2: A[1,2](m)= m;
n=3: A[1,3](m)= m/2 + m^2;
n=4: A[1,4](m)= m/4 + m^2/3 + m^3/2 + m^4.
General formula which uses these polynomials is:
(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] = Sum_{x>=0} m^(-x)/(x+n) = m^n*arctanh((2m-1)/(2m^2-2m+1)) - A[1,n](m) = m^n*log(m/(m-1)) - A[1,n](m).
The sequence of denominators is ?, 1, 2, 6, 12, 12, 12, 84, ... - Matthew J. Samuel, Jan 30 2011

Cf. A145609-A145640, A145656, A145658, A145660, A145664, A145666.

Cf. A006245.

A145662 := proc(n) add( 5^(n-d)/d,d=1..n-1) ; numer(%) ; end proc: # R. J. Mathar, Feb 01 2011

m = 5; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa

A145664 a(n) = numerator of polynomial of genus 1 and level n for m = 6 = A[1,n](6).

Original entry on oeis.org

0, 6, 39, 236, 2835, 42531, 255191, 10718052, 257233353, 2315100317, 2315100338, 152796622518, 1833559470601, 71508819355749, 429052916136639, 2574317496821836, 123567239847463143, 6301929232220740413

Offset: 1

Artur Jasinski, Oct 16 2008

For numerator of polynomial of genus 1 and level n for m = 1 see A001008.
Definition: The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as
Sum_{d=1..n-1} m^(n-d)/d.
Few first A[1,n](m):
n=1: A[1,1](m)= 0;
n=2: A[1,2](m)= m;
n=3: A[1,3](m)= m/2 + m^2;
n=4: A[1,4](m)= m/4 + m^2/3 + m^3/2 + m^4;
General formula which uses these polynomials is:
(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =
Sum_{x>=0} m^(-x)/(x+n) =
m^n*arctanh((2m-1)/(2m^2-2m+1)) - A[1,n](m) =
m^n*log(m/(m-1)) - A[1,n](m).

Cf. A145609-A145640, A145656, A145658, A145660, A145662, A145664.

A145664 := proc(n) add( 6^(n-d)/d,d=1..n-1) ; numer(%) ; end proc:
seq(A145664(n),n=1..20) ; # R. J. Mathar, Feb 01 2011

m = 6; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa

A145666 a(n) = numerator of polynomial of genus 1 and level n for m = 7 : A[1,n](7).

Original entry on oeis.org

0, 7, 105, 2219, 31087, 1088129, 2538991, 17772957, 248821433, 15675750559, 21946050833, 1689845914645, 11828921402977, 1076431847676451, 7535022933740305, 263725802680934699, 3692161237533130831

Offset: 1

Artur Jasinski, Oct 16 2008

For numerator of polynomial of genus 1 and level n for m = 1 see A001008.
Definition: The polynomial A[1,n](m) = A[genus 1,level n] is here defined as
Sum_{d=1..n-1} m^(n - d)/d
Few first A[1,n](m):
n=1: A[1,1](m)= 0;
n=2: A[1,2](m)= m;
n=3: A[1,3](m)= m/2 + m^2;
n=4: A[1,4](m)= m/3 + m^2/2 + m^3;
n=5: A[1,5](m)= m/4 + m^2/3 + m^3/2 + m^4.
General formula which uses these polynomials is:
(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =
Sum_{x>=0} m^(-x)/(x+n) =
m^(n)*arctanh((2m-1)/(2m^2-2m+1)) - A[1,n](m) =
m^(n)*log(m/(m-1)) - A[1,n](m).

Cf. A145609-A145640, A145656-A145687.

A145666 := proc(n) add( 7^(n-d)/d,d=1..n-1) ; numer(%) ; end proc:
seq(A145666(n),n=1..20) ; # R. J. Mathar, Feb 01 2011

m = 7; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa

A145611 Numerator of the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=2.

Original entry on oeis.org

5, 131, 1327, 148969, 89422, 7869871, 204620705, 32739453941, 556571247527, 42299423848079, 84598851790183, 31132377803126339, 155661889412050564, 3735885348093583561, 216681350219210744683, 429895798848743086730197

Offset: 1

Artur Jasinski, Oct 14 2008

For denominators see A145612. For general properties of A_l(x) see A145609.

Cf. A145609 - A145640.

A := proc(l,x) add(x^(l-d)/d,d=1..l-1) ; end: A145611 := proc(n) numer( A(2*n+1,2)) ; end: seq(A145611(n),n=1..20) ; # R. J. Mathar, Aug 21 2009

m = 2; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Numerator[k]], {r, 1, 25}]; aa (* Artur Jasinski, Oct 14 2008 *)
a[n_,m_]:=Integrate[(m-x^n)/(m-x),{x,0,1}]+(m^n-m)Log[m/(m-1)]
Table[2 a[2 n, 2] // Simplify  // Numerator, {n,1,25}]  (* Gerry Martens , Jun 04 2016 *)

Edited by R. J. Mathar, Aug 21 2009

cp's OEIS Frontend

A145737 a(n) = square part of A145609(n).

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A145738 a(n) = squarefree part of A145609(n).

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A001008 a(n) = numerator of harmonic number H(n) = Sum_{i=1..n} 1/i.

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A145640 Denominator the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=16.

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A145656 a(n) = numerator of polynomial of genus 1 and level n for m = 2.

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A145660 a(n) = numerator of polynomial of genus 1 and level n for m = 4 = A[1,n](4).

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A145662 a(n) = numerator of polynomial of genus 1 and level n for m = 5 = A[1,n](5).

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A145664 a(n) = numerator of polynomial of genus 1 and level n for m = 6 = A[1,n](6).

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