A025547 -id:A025547 - cp's OEIS frontend

A092318 a(n) = smallest m such that value of odd harmonic series Sum_{j=0..m} 1/(2j+1) is >= n.

0, 7, 56, 418, 3091, 22845, 168803, 1247297, 9216353, 68100150, 503195828, 3718142207, 27473561357, 203003686105, 1500005624923, 11083625711270, 81897532160124, 605145459495140, 4471453748222756, 33039822589391675

Offset: 1

N. J. A. Sloane, Feb 16 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (127 terms corrected by Gerhard Kirchner)

Apart from first term, same as A092315. Equals (A092317-1)/2.
Cf. A074599, A025547.
Cf. A281355 (= a(n) + 1) for a variant.

a[n_] := Floor[(Exp[2 n - EulerGamma] + 1/2)/4]; a[1] = 0; Array[a, 20] (* Robert G. Wilson v, Jan 25 2017 *)

A092318=n->floor(exp(2*n-Euler)/4+1/8)-(n<2) \\ Cf. comments in A092315. - M. F. Hasler, Jan 24 2017

a(n) = floor(exp(2*n-gamma)/4+1/8), for all n > 1. - M. F. Hasler and Robert G. Wilson v, Jan 22 2017

a(n) = floor(exp(2*n-gamma)/4), for all n > 1, see correction in A092315, Gerhard Kirchner, Jul 25 2020

More terms (computed from A092317) from M. F. Hasler, Jan 22 2017

a(17) corrected by Gerhard Kirchner, Jul 26 2020

A350669 Numerators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 16 2022

For the denominators see A350670.
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.

Hugo Pfoertner, Table of n, a(n) for n = 0..100
Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. p.258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. p. 258.
Comparison to A025550 using Plot 2.

Cf. A001620, A025547, A025550, A111877 (denominators), A350670.

[Numerator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1)))/2: n in [0..40]]; // G. C. Greubel, Jul 24 2023

With[{H=HarmonicNumber}, Table[Numerator[2*H[2*n+2] -H[n+1]]/2 , {n,0,50}]] (* G. C. Greubel, Jul 24 2023 *)

a(n) = numerator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022

[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

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

A350670 Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 16 2022

For the numerators see A350669.
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.

Hugo Pfoertner, Table of n, a(n) for n = 0..100
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions. p. 258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 258.
Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
Comparison to A025547 using Plot 2.

Cf. A001620, A025547, A025550, A350669 (numerators).

[Denominator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1))): n in [0..40]]; // G. C. Greubel, Jul 24 2023

With[{H=HarmonicNumber}, Table[Denominator[2*H[2n+2] -H[n+1]], {n,0,50}]] (* G. C. Greubel, Jul 24 2023 *)

a(n) = denominator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022

[denominator(2*harmonic_number(2*n+2,1) - harmonic_number(n+1,1)) for n in range(41)] # G. C. Greubel, Jul 24 2023

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

A025549 a(n) = (2n-1)!!/lcm{1,3,5,...,2n-1}.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 45, 45, 45, 945, 945, 4725, 42525, 42525, 42525, 1403325, 49116375, 49116375, 1915538625, 1915538625, 1915538625, 86199238125, 86199238125, 603394666875, 30773128010625, 30773128010625, 1692522040584375, 96473756313309375, 96473756313309375

Offset: 1

Clark Kimberling

Michael De Vlieger, Table of n, a(n) for n = 1..569

Not always equal to the second left hand column of A161198 triangle divided by A074599. - Johannes W. Meijer, Jun 08 2009

Cf. A196274 (run lengths of equal terms).

seq(doublefactorial(2*n-1)/lcm(seq((2*k-1), k=1..n)), n=1..27) ; # Johannes W. Meijer, Jun 08 2009

L[ {x___} ] := LCM[ x ]; Table[ (2n-1)!!/L[ Range[ 1, 2n-1, 2 ] ], {n, 1, 50} ]
(* Second program: *)
Array[#!!/LCM @@ Range[1, #, 2] &[2 # - 1] &, 30] (* Michael De Vlieger, Feb 19 2019 *)

a(n) = (((2*n)!/n!)/2^n)/lcm(vector(n, i, 2*i-1)); \\ Michel Marcus, Dec 02 2014

a(n) = A001147(n)/A025547(n). - Michel Marcus, Dec 02 2014

Description corrected and sequence extended by Erich Friedman

More terms from Michel Marcus, Dec 02 2014

A167577 The second column of the ED3 array A167572.

Original entry on oeis.org

1, 11, 83, 741, 8169, 106107, 1592235, 27062325, 514246545, 10798366635, 248374594755, 6209158112325, 167651197407225, 4861802228946075, 150717766502187675, 4973638859450709525, 174078640829054894625

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..150

Equals the second column of the ED3 array A167572.
Other columns are A167576 and A167578.
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).

Table[(1/2)*(-1)^n*(2*n - 5)!!*((4*n^2 - 6*n - 2) + (16*n^3 - 24*n^2 - 4*n + 6)*Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}]), {n, 1,50}] (* G. C. Greubel, Jun 16 2016 *)

a(n) = (1/2)*(-1)^n*(2*n-5)!!*((4*n^2-6*n-2)+(16*n^3-24*n^2-4*n+6)*sum((-1)^(k+n)/ (2*k+1), k=0..n-1)).

A167578 The third column of the ED3 array A167572.

Original entry on oeis.org

1, 17, 183, 2043, 26529, 398025, 6765975, 128556675, 2699661825, 62092533825, 1552309291575, 41912411683275, 1215458905032225, 37679245697871225, 1243414695550433175, 43519523831289457875, 1610222144582102522625

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..150

Equals the third column of the ED3 array A167572.
Other columns are A167576 and A167577.
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).

Table[(1/4)*(-1)^(n)*(2*n - 7)!!*((8*n^4 - 20*n^3 - 22*n^2 + 55*n + 12) + (32*n^5 - 80*n^4 - 80*n^3 + 200*n^2 + 18*n - 45)*(Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}])), {n, 1, 50}] (* G. C. Greubel, Jun 16 2016 *)

a(n) = (1/4)*(-1)^(n)*(2*n-7)!!*((8*n^4-20*n^3-22*n^2+55*n+12)+(32*n^5-80*n^4-80*n^3+200*n^2+18*n-45)*(sum((-1)^(k+n)/(2*k+1),k=0..n-1))).

A051426 Least common multiple of {2, 4, 6, ..., 2n}.

Original entry on oeis.org

2, 4, 12, 24, 120, 120, 840, 1680, 5040, 5040, 55440, 55440, 720720, 720720, 720720, 1441440, 24504480, 24504480, 465585120, 465585120, 465585120, 465585120, 10708457760, 10708457760, 53542288800, 53542288800, 160626866400, 160626866400, 4658179125600

Offset: 1

Asher Auel

Gcd(A025547(n), a(n)) = A025547(floor((n+1)/2)). - Reinhard Zumkeller, Apr 25 2011

			a(3) = lcm{2,4,6} = 12;
a(7) = lcm{2,4,6,8,10,12,14} = 840.

A. Murthy, Some new Smarandache sequences, functions and partitions, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000 (but beware errors).

Cf. A003418, A025547.

a051426 n = foldl lcm 1 [2,4..2*n] -- Reinhard Zumkeller, Apr 25 2011

SZ={2};n=2;L=2;Do[L=LCM[L,2n];AppendTo[SZ,L];n++,{99}];SZ (* Zak Seidov, Aug 01 2009 *)
Table[LCM@@Range[2,2n,2],{n,30}] (* Harvey P. Dale, Oct 09 2011 *)

a(n)=2*lcm([1..n]) \\ Charles R Greathouse IV, Oct 28 2016

a(n) = a(n-1)*lcm(a(n-1),2n) = a(n-1)* A014963(n). - Zak Seidov, Aug 01 2009

a(6), a(7) and a(8) corrected by T. D. Noe, Feb 08 2008
Corrected the example, which did not reflect the sequence values provided. - Michael Davies (mykdavies+oeis(AT)gmail.com), Oct 10 2008
Edited by N. J. A. Sloane, Jul 31 2009 at the suggestion of R. J. Mathar

A092317 a(n) = smallest odd number 2m+1 such that the partial sum Sum_{j=0..m} 1/(2j+1) of the odd harmonic series is >= n.

Original entry on oeis.org

1, 15, 113, 837, 6183, 45691, 337607, 2494595, 18432707, 136200301, 1006391657, 7436284415, 54947122715, 406007372211, 3000011249847, 22167251422541, 163795064320249, 1210290918990281, 8942907496445513, 66079645178783351

Offset: 1

N. J. A. Sloane, Feb 16 2004

nonn

Except for first term, same as A056053. Equals 2*A092318 + 1. Cf. A074599, A025547

a(n) ~ C*exp(2n) with C = 0.2807297417834425... - M. F. Hasler, Jan 22 2017

More terms (via A056053) from M. F. Hasler, Jan 22 2017

a(17) corrected - see correction in A092315. Gerhard Kirchner, Jul 25 2020

A164655 Numerators of partial sums of Theta(3) = Sum_{j>=1} 1/(2*j-1)^3.

Original entry on oeis.org

1, 28, 3527, 1213136, 32797547, 43684790932, 96017087247229, 96044168328256, 471956397645187853, 3237597973008257555852, 462561506842656976961, 5628425850334528955928112, 703596058798919360293439483, 18998011529681231695738912916, 463360571051954739540899597748949

Offset: 1

Wolfdieter Lang, Oct 16 2009

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.

			Rationals Theta(3,n): [1, 28/27, 3527/3375, 1213136/1157625, 32797547/31255875, 43684790932/41601569625, ...].

R. Ayoub, Euler and the Zeta Function, Am. Math. Monthly 81 (1974) 1067-1086, p. 1070.
Wolfdieter Lang, Theta(k, n), k-family of rational sequences and limits.

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 *)

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

A164656 Numerators of partial sums of Theta(5) = sum( 1/(2*j-1)^5, j=1..infinity ).

Original entry on oeis.org

1, 244, 762743, 12820180976, 3115356499043, 501734380891571068, 186290962962179367466549, 186291207179611798681792, 264507060005034822095008296869, 654945930087597102815813733559637156, 654946089730308117005814730177159031, 4215458332009996232497953858159263996273008

Offset: 1

Wolfdieter Lang, Oct 16 2009

The denominators are given by A164657.
Rationals (partial sums) Theta(5,n) := sum(1/(2*j-1)^5,j=1..n) (in lowest terms). The limit of these rationals is Theta(5)= (1-1/2^5)*Zeta(5) approximately 1.004523763.., see A013663.
This is a member of the k-family of rational sequences Theta(k,n):=sum(1/(2*j-1)^k,j=1..n), k>=1, which includes A025550/A025547 (but only for the first 38 entries), A120268/A128492, A164655(n)/A128507(n) (the denominators may depart for higher n values), A120269/A128493, a(n)/A164657, for k=1..5.

			Rationals Theta(5,n): [1, 244/243, 762743/759375, 12820180976/12762815625, 3115356499043/3101364196875,...].

R. Ayoub, Euler and the Zeta Function, Am. Math. Monthly 81 (1974) 1067-1086.
W. Lang: Theta(k,n), k-family of rational sequences and limits.

r[n_] := Sum[1/(2*j-1)^5, {j, 1, n}]; (* or r[n_] := (PolyGamma[4, n+1/2] - PolyGamma[4, 1/2])/768 // FullSimplify; *) Table[r[n] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 02 2013 *)

a(n) = numer(Theta(5,n))= numerator(sum(1/(2*j-1)^5,j=1..n)), n>=1.

Theta(5,n) = (-Psi(4, 1/2) + Psi(4, n+1/2))/(4!*2^5), n >= 1, with Psi(n,k) = Polygamma(n,k) is the n^th derivative of the digamma function. Psi(4, 1/2) = -4!*31*Zeta(5). - Jean-François Alcover, Dec 02 2013

cp's OEIS Frontend

A092318 a(n) = smallest m such that value of odd harmonic series Sum_{j=0..m} 1/(2j+1) is >= n.

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A350669 Numerators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

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A350670 Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

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A025549 a(n) = (2n-1)!!/lcm{1,3,5,...,2n-1}.

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A167577 The second column of the ED3 array A167572.

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A167578 The third column of the ED3 array A167572.

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A051426 Least common multiple of {2, 4, 6, ..., 2n}.

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