Displaying 1-10 of 11 results found.
Numerator of 2 * H(n,2,1), a generalized harmonic number. See A075135. Also 2 * A025550.
+20
9
2, 8, 46, 352, 1126, 13016, 176138, 182144, 3186538, 62075752, 63461422, 1488711776, 7577414602, 23104065256, 680057071574, 21372905414144, 21646396991594, 21904260478904, 819482859775298, 828045249930848
MATHEMATICA
Table[ Numerator[ Sum[1/i, {i, 1/2, n}]], {n, 1, 20}]
CROSSREFS
Not always equal to the second left hand column of A161198 triangle divided by A025549. (End)
Least common multiple of {1,3,5,...,2n-1}.
+10
35
1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675
COMMENTS
This sequence coincides with the sequence f(n) = denominator of 1 + 1/3 + 1/5 + 1/7 + ... + 1/(2n-1) iff n <= 38. But a(39) = 6414924694381721303722858446525, f(39) = 583174972216520118520259858775. - T. D. Noe, Aug 04 2004 [See A350670(n-1).] Coincides for n=1..42 with the denominators of a series for Pi*sqrt(2)/4 and then starts to differ. See A127676.
MAPLE
A025547:=proc(n) local i, t1; t1:=1; for i from 1 to n do t1:=lcm(t1, 2*i-1); od: t1; end;
f := n->denom(add(1/(2*k-1), k=0..n)); # a different sequence!
MATHEMATICA
a = 1; Join[{1}, Table[a = LCM[a, n], {n, 3, 125, 2}]] (* Zak Seidov, Jan 18 2011 *) nn=30; With[{c=Range[1, 2*nn, 2]}, Table[LCM@@Take[c, n], {n, nn}]] (* Harvey P. Dale, Jan 27 2013 *)
PROG
(Haskell)
a025547 n = a025547_list !! (n-1)
a025547_list = scanl1 lcm a005408_list
(Python) # generates initial segment of sequence
from math import gcd
from itertools import accumulate
def lcm(a, b): return a * b // gcd(a, b)
def aupton(nn): return list(accumulate((2*i+1 for i in range(nn)), lcm))
Numerator of the generalized harmonic number H(n,3,1) described below.
+10
25
1, 5, 39, 209, 2857, 11883, 233057, 2632787, 13468239, 13739939, 433545709, 7488194853, 281072414761, 284780929571, 12393920563953, 288249495707519, 2038704876507433, 2058454144222533, 2077126179153173, 60750140156034617
COMMENTS
For integers a and b, H(n,a,b) is the sum of the fractions 1/(a i + b), i = 0,1,..,n-1. This database already contains six instances of generalized harmonic numbers. Partial sums of the harmonic series 1+1/2+1/3+1/4+... are given by the sequence of harmonic numbers H(n,1,1) = A001008(n) / A002805(n). The Jeep problem gives rise to the series H(n,2,1) = A025550(n) / A025547(n). Recent additions to the database are 3 * H(n,3,1) = A074596(n) / A051536(n), 3 * H(n,3,2) = A074597(n) / A051540(n), 4 * H(n,4,1) = A074598(n) / A051539(n) and 4 * H(n,4,3) = A074637(n) / A074638(n) . The numerator of H(n,4,1) is A075136. The fractions H(n,5,1), H(n,5,2), H(n,5,3) and H(n,5,4) are in A075137- A075144. The sequence H(n,a,b) is of interest only when a and b are relatively prime. The sequence can also be computed as H(n,a,b) = (PolyGamma[n+1+b/a] - PolyGamma[1+b/a])/a. The sequence H(n,a,b) diverges for all a and b.
According to Hardy and Wright, if p is an odd prime, then p divides the numerator of the harmonic number H(p-1,1,1). This result can be extended to generalized harmonic numbers: for odd integer n, let q = (n-2)a + 2b. If q is prime, then q divides the numerator of H(n-1,a,b). For this sequence (a=3, b=1) we conclude that 11 divides a(4), 17 divides a(6), 29 divides a(10) and 47 divides a(16).
Graham, Knuth and Patashnik define another type of generalized harmonic number as the sum of fractions 1/i^k, i=1,...,n. For k=2, the sequence of fractions is A007406(n) / A007407(n).
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 263, 269, 272, 297, 302, 356.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 88.
EXAMPLE
a(3)=39 because 1 + 1/4 + 1/7 = 39/28.
MATHEMATICA
a=3; b=1; maxN=20; s=0; Numerator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]
Accumulate[1/Range[1, 60, 3]]//Numerator (* Harvey P. Dale, Dec 30 2019 *)
Numerator of Sum_{k=1..n} 1/(2*k-1)^2.
+10
18
1, 10, 259, 12916, 117469, 14312974, 2430898831, 487983368, 141433003757, 51174593563322, 51270597630767, 27164483940418988, 3400039831130408821, 30634921277843705014, 25789165074168004597399
COMMENTS
a((p-1)/2) is divisible by prime p > 3.
The limit of the rationals r(n) = Sum_{k=1..n} 1/(2*k-1)^2, for n -> infinity, is (Pi^2)/8 = (1 - 1/2^2)*Zeta(2), which is approximately 1.233700550.
r(n) = (Psi(1, 1/2) - Psi(1, n+1/2))/4 for n >= 1, where Psi(n,k) = Polygamma(n,k) is the n-th derivative of the digamma function. Psi(1, 1/2) = 3*Zeta(2) = Pi^2/2. - Jean-François Alcover, Dec 02 2013 [Corrected by Petros Hadjicostas, May 09 2020]
EXAMPLE
Fractions begin: 1, 10/9, 259/225, 12916/11025, 117469/99225, 14312974/12006225, 2430898831/2029052025, 487983368/405810405, ... = A120268/ A128492.
MATHEMATICA
Numerator[Table[Sum[1/(2k-1)^2, {k, 1, n}], {n, 1, 25}]]
Table[(PolyGamma[1, 1/2] - PolyGamma[1, n+1/2])/4 // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *) Accumulate[1/(2*Range[20]-1)^2]//Numerator (* Harvey P. Dale, Jun 14 2020 *)
PROG
(PARI) for(n=1, 20, print1(numerator(sum(k=1, n, 1/(2*k-1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018 (Magma) [Numerator((&+[1/(2*k-1)^2: k in [1..n]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018
T(j,k) are the numerators s in the representation R = s/t + (2/Pi)*u/v of the resistance between two nodes separated by the distance vector (j,k) in an infinite square lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= j, is a triangle read by rows.
+10
10
0, 1, 0, 2, -1, 0, 17, -4, 1, 0, 40, -49, 6, -1, 0, 401, -140, 97, -8, 1, 0, 1042, -1569, 336, -161, 10, -1, 0, 11073, -4376, 4321, -660, 241, -12, 1, 0, 29856, -48833, 13342, -9681, 1144, -337, 14, -1, 0, 325441, -136488, 160929, -33188, 18929, -1820, 449, -16, 1, 0
COMMENTS
The recurrence given by Cserti (2000), page 5, (32) is used to calculate the resistance between two arbitrarily spaced nodes in an infinite square lattice whose edges are replaced by one-ohm resistors. The lower triangle, including the diagonal, in Table I of Atkinson and Steenwijk (1999), page 487, is reproduced. The solution to the resistor grid problem shown in the xkcd Web Comic #356 "Nerd Sniping", provided in A211074, is the special case (j,k) = (2,1). Using the terms of A280079 and A280317 as pairs of grid indices leads to strictly increasing resistances, i.e., R( A280079(m), A280317(m)) > R( A280079(i), A280317(i)) for m > i. This implies that for grid points on the same radius the resistance increases with the circumferential angle between 0 and Pi/4. The further dependence of the resistance along the circumferential angle with a fixed radius results from symmetry. - Hugo Pfoertner, Aug 31 2022
REFERENCES
See A211074 for more references and links.
LINKS
Hugo Pfoertner, PARI program for inverse problem, (2022). Finds the grid point [x,y] that leads to the best approximation of a given resistance distance R (ohms) between [0,0] and [x,y].
FORMULA
The resistance for the distance vector (j,k) is R(j,k) = T(j,k)/(1+mod(j+k,2)) +(2/Pi)* A355566(j,k)/ A355567(j,k), avoiding the use of A131406. R(0,0) = 0; R(1,0) = 1/2.
R(n,n) = R(n-1,n-1) + (2/Pi)/(2*n-1) for n >= 1.
R(j,k) = R(k,j) and R(-j,k) = R(j,k).
4*R(j,k) = R(j-1,k) + R(j+1,k) + R(j,k-1) + R(j,k+1) for (j,k) != (0,0).
(End)
EXAMPLE
The triangle begins:
0;
1, 0;
2, -1, 0;
17, -4, 1, 0;
40, -49, 6, -1, 0;
401, -140, 97, -8, 1, 0;
1042, -1569, 336, -161, 10, -1, 0
.
The combined triangles used to calculate the resistances are:
\ k 0 | 1 | 2 | 3 |
\ s/t u/v | s/t u/v | s/t u/v | s/t u/v |
j \---------------|-----------------|---------------|--------------|
0 | 0 0 | . . | . . | . . |
1 | 1/2 0 | 0 1 | . . | . . |
2 | 2 -2 | -1/2 2 | 0 4/3 | . . |
3 | 17/2 -12 | -4 23/3 | 1/2 2/3 | 0 23/15 |
4 | 40 -184/3 | - 49/2 40 | 6 -118/15 | -1/2 12/5 |
5 | 401/2 -940/3 | -140 3323/15 | 97/2 -1118/15 | -8 499/35 |
.
continued:
\ k 4 | 5 |
\ s/t u/v | s/t u/v |
j \-------------|--------------|
0 | . . | . . |
1 | . . | . . |
2 | . . | . . |
3 | . . | . . |
4 | 0 176/105 | . . |
5 | 1/2 20/21 | 0 563/315 |
.
E.g., the resistance for a node distance vector (4,1) is R = T(4,1)/ A131406(5,2) + (2/Pi)* A355566(4,1)/ A355567(4,1) = -49/2 + (2/Pi)*40/1 = 80/Pi - 49/2.
MATHEMATICA
alphas[beta_] :=
Log[2 - Cos[beta] + Sqrt[3 + Cos[beta]*(Cos[beta] - 4)]];
Rsqu[n_, p_] :=
Simplify[(1/Pi)*
Integrate[(1 - Exp[-Abs[n]*alphas[beta]]*Cos[p*beta])/
Sinh[alphas[beta]], {beta, 0, Pi}]];
Table[Rsqu[n, k], {n, 0, 4}, {k, 0, n}] // TableForm (* Hugo Pfoertner, Aug 21 2022, calculates R, after Atkinson and Steenwijk *)
PROG
(PARI) R(m, p, x=pi) = {if (m==0 && p==0, return(0)); if (m==1 && p==0, return(1/2)); if (m==1 && p==1, return(2/x)); if(m==p, my(mm=m-1); return(R(mm, mm)*4*mm/(2*mm+1) - R(mm-1, mm-1)*(2*mm-1)/(2*mm+1))); if (p==(m-1), my(mm=m-1); return(2*R(mm, mm) - R(mm, mm-1))); if (p==0, my(mm=m-1); return(4*R(mm, 0) - R(mm-1, 0) - 2*R(mm, 1))); if (p<m && p>0, my(mm=m-1); return(4*R(mm, p) - R(mm-1, p) - R(mm, p+1) - R(mm, p-1)))};
for(j=0, 9, for(k=0, j, my(q=pi*R(j, k, pi)); print1(numerator(polcoef(q, 1, pi)), ", ")); print())
CROSSREFS
A131406 are the corresponding denominators t, with indices shifted by 1.
Numerators of coefficients of expansion of arctan(x)^2 = x^2 - 2/3*x^4 + 23/45*x^6 - 44/105*x^8 + 563/1575*x^10 - 3254/10395*x^12 + ...
(Formerly M2131 N0844)
+10
7
0, 1, -2, 23, -44, 563, -3254, 88069, -11384, 1593269, -15518938, 31730711, -186088972, 3788707301, -5776016314, 340028535787, -667903294192, 10823198495797, -5476065119726, 409741429887649, -103505656241356, 17141894231615609
COMMENTS
|a(n)| = numerator of Sum_{k=1..n} 1/(n*(2*k-1)).
Let f(x) = (1/2)*log((1+sqrt(x))/(1-sqrt(x))) and c(n) = Integral_{x=0..1} f(x)*x^(n-1) dx, then for n>=1, c(n) = |a(n+1)|/ A071968(n) and (f(x))^2 = Sum_{n>=1} c(n)*x^n. - Groux Roland, Dec 14 2010
REFERENCES
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 89.
H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 313.
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).
FORMULA
a(n) = numerator of (-1)^n * Sum_{k=1..n-1} 1/((n-1)*(2*k-1)), for n>=1. - G. C. Greubel, Jul 03 2019
MATHEMATICA
a[n_]:= (-1)^n*Sum[1/((n-1)*(2*k-1)), {k, 1, n-1}]//Numerator; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 04 2013 *) a[n_]:= SeriesCoefficient[ArcTan[x]^2, {x, 0, 2*n-2}]//Numerator; Table[a[n], {n, 1, 30}] (* G. C. Greubel, Jul 03 2019 *)
PROG
(PARI) vector(30, n, numerator((-1)^n*sum(k=1, n-1, 1/((n-1)*(2*k-1))))) /* corrected by G. C. Greubel, Jul 03 2019 */ (Magma) [0] cat [Numerator((-1)^n*(&+[1/((n-1)*(2*k-1)): k in [1..n-1]])): n in [2..30]]; // G. C. Greubel, Jul 03 2019 (Sage) [numerator((-1)^n*sum(1/((n-1)*(2*k-1)) for k in (1..n-1))) for n in (1..30)] # G. C. Greubel, Jul 03 2019 (GAP) List([1..30], n-> NumeratorRat( (-1)^n*Sum([1..n-1], k-> 1/((n-1)*(2*k-1))) )) # G. C. Greubel, Jul 03 2019
Numerators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.
+10
6
1, 4, 23, 176, 563, 6508, 88069, 91072, 1593269, 31037876, 31730711, 744355888, 3788707301, 11552032628, 340028535787, 10686452707072, 10823198495797, 10952130239452, 409741429887649, 414022624965424, 17141894231615609, 743947082888833412, 750488463554668427, 35567319917031991744, 250947670863258378883, 252846595191840484708, 13497714685925233086599
COMMENTS
This sequence coincides with A025550(n+1), for n = 0, 1, ..., 37. See the comments there. Thanks to Ralf Steiner for sending me a paper where this and similar sums appear.
FORMULA
a(n) = numerator((Psi(n+3/2) + gamma + 2*log(2))/2), with the Digamma function Psi(z), and the Euler-Mascheroni constant gamma = A001620. See Abramowitz-Stegun, p. 258. 6.3.4. a(n) = (1/2) * numerator of ( 2*H_{2*n+2} - H_{n+1} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023
MATHEMATICA
With[{H=HarmonicNumber}, Table[Numerator[2*H[2*n+2] -H[n+1]]/2 , {n, 0, 50}]] (* G. C. Greubel, Jul 24 2023 *)
PROG
(PARI) a(n) = numerator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022 (Magma) [Numerator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1)))/2: n in [0..40]]; // G. C. Greubel, Jul 24 2023 (SageMath) [numerator(2*harmonic_number(2*n+2, 1) - harmonic_number(n+1, 1))/2 for n in range(41)] # G. C. Greubel, Jul 24 2023
Numerator of Sum_{k=1..n} 1/(2k-1)^4.
+10
5
1, 82, 51331, 123296356, 9988505461, 146251554055126, 4177234784807204311, 4177316109293528392, 348897735816424941428857, 45469045689642442391390873722, 45469276109166591994111574347
COMMENTS
a((p-1)/2) is divisible by prime p > 5.
The limit of the rationals r(n) = Sum_{k=1..n} 1/(2k-1)^4, for n -> infinity, is (Pi^4)/96 = (1 - 1/2^4)*zeta(4), which is approximately 1.014678032.
r(n) = (Psi(3, 1/2) - Psi(3, n+1/2))/(3!*2^4) for n >= 1, where Psi(n,k) = Polygamma(n,k) is the n^th derivative of the digamma function. Psi(3, 1/2) = 3!*15*zeta(4) = Pi^4. - Jean-François Alcover, Dec 02 2013
MATHEMATICA
Numerator[Table[Sum[1/(2k-1)^4, {k, 1, n}], {n, 1, 20}]]
Table[(PolyGamma[3, 1/2] - PolyGamma[3, n + 1/2])/(3!*2^4) // Simplify // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *)
PROG
(PARI) for(n=1, 20, print1(numerator(sum(k=1, n, 1/(2*k-1)^4)), ", ")) \\ G. C. Greubel, Aug 23 2018 (Magma) [Numerator((&+[1/(2*k-1)^4: k in [1..n]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018
Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.
+10
5
1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725, 96845140757687397075, 96845140757687397075, 5132792460157432044975
COMMENTS
This sequence coincides with A025547(n+1), for n = 0, 1, ..., 37. See the comments there. Thanks to Ralf Steiner for sending me a paper where this and similar sums appear.
FORMULA
a(n) = denominator((Psi(n+3/2) + gamma + 2*log(2))/2), with the Digamma function Psi(z), and the Euler-Mascheroni constant gamma = A001620. See Abramowitz-Stegun, p. 258, 6.3.4. a(n) = denominator of ( 2*H_{2*n+2} - H_{n+1} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023
MATHEMATICA
With[{H=HarmonicNumber}, Table[Denominator[2*H[2n+2] -H[n+1]], {n, 0, 50}]] (* G. C. Greubel, Jul 24 2023 *)
PROG
(PARI) a(n) = denominator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022 (Magma) [Denominator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1))): n in [0..40]]; // G. C. Greubel, Jul 24 2023 (SageMath) [denominator(2*harmonic_number(2*n+2, 1) - harmonic_number(n+1, 1)) for n in range(41)] # G. C. Greubel, Jul 24 2023
Numerators of partial sums of Theta(3) = Sum_{j>=1} 1/(2*j-1)^3.
+10
4
1, 28, 3527, 1213136, 32797547, 43684790932, 96017087247229, 96044168328256, 471956397645187853, 3237597973008257555852, 462561506842656976961, 5628425850334528955928112, 703596058798919360293439483, 18998011529681231695738912916, 463360571051954739540899597748949
COMMENTS
Warning: Usually, Theta3(x) = Sum_{n=-oo..+oo} x^(n^2). - Joerg Arndt, Mar 31 2024 The denominators look like those given for the partial sums of another series in A128507. Rationals (partial sums) Theta(3,n) := Sum_{j=1..n} 1/(2*j-1)^3 (in lowest terms). The limit of these rationals is Theta(3) = (1-1/2^3)*Zeta(3) approximately 1.051799790 (Zeta(n) is the Euler-Riemann zeta function).
This is a member of the k-family of rational sequences Theta(k,n) := Sum_{j=1..n} 1/(2*j-1)^k, k >= 1, which coincides for k=1 with A025550/ A025547 (but only for the first 38 terms), for k=2 with A120268/ A128492, for k=3 with a(n)/ A128507(n) (the denominators may depart for higher n values), A120269/ A128493 and A164656/ A164657, for k=4 and 5, respectively.
FORMULA
a(n) = numerator(Theta(3,n)) = numerator(Sum_{j=1..n} 1/(2*j-1)^3), n >= 1.
Theta(3,n) = (-Psi(2, 1/2) + Psi(2, n+1/2))/16, n >= 1, where Psi(n, k) = Polygamma(n,k) is the n-th derivative of the digamma function. Psi(2, 1/2) = -14*Zeta(3). - Jean-François Alcover, Dec 02 2013
EXAMPLE
Rationals Theta(3,n): [1, 28/27, 3527/3375, 1213136/1157625, 32797547/31255875, 43684790932/41601569625, ...].
MATHEMATICA
r[n_] := Sum[1/(2*j-1)^3, {j, 1, n}]; (* or r[n_] := (PolyGamma[2, n+1/2] - PolyGamma[2, 1/2])/16 // FullSimplify; *) Table[r[n] // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *)
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