A050971 -id:A050971 - cp's OEIS frontend

A050970 Numerator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} (4k+1)^(-n).

1, 1, 1, 1, 5, 1, 61, 17, 277, 31, 50521, 691, 540553, 5461, 199360981, 929569, 3878302429, 3202291, 2404879675441, 221930581, 14814847529501, 4722116521, 69348874393137901, 56963745931, 238685140977801337, 14717667114151

Offset: 1

Eric W. Weisstein

Reduced numerators of Favard constants.

			The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..200
N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001.
N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
Z. K. Silagadze, Comment on the sums S(n) = sum(k=-inf..inf) 1/(4k+1)^n, (2012) arXiv:1207.2055
Eric Weisstein's World of Mathematics, Favard Constants

Denominators: A068205. See also A050971.

Cf. A000828, A046982.

S := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else
S(n, k - 1) + S(n - 1, n - k) fi end: EZ := n -> S(n, n)/(2^n * n!):
A050970 := n -> numer(EZ(n-1)): seq(A050970(n), n=1..26); # Peter Luschny, Aug 02 2017
# alternative
A050970 := proc(n)
    if type(n,'even') then
        (-1)^(n/2)*2^(n-2)/(n-1)!*euler(n-1,0) ;
    else
        (-1)^((n-1)/2)*2^(n-2)/(n-1)!*euler(n-1,1/2) ;
    end if;
    %/2^n ;
    numer(%) ;
end proc:
seq(A050970(n),n=1..20) ; # R. J. Mathar, Jun 26 2024

s[n_] := Sum[(4*k + 1)^(-n), {k, -Infinity, Infinity}]; a[n_] := Numerator[FullSimplify[s[n]/Pi^n]]; a[1] = 1; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 25 2012 *)
s[n_?EvenQ] := (-1)^(n/2-1)*(2^n-1)*BernoulliB[n]/(2*n!); s[n_?OddQ] := (-1)^((n-1)/2)*2^(-n-1)*EulerE[n-1]/(n-1)!; Table[s[n] // Numerator, {n, 1, 26}] (* Jean-François Alcover, May 13 2013 *)
a[n_] := 4*Sum[((-1)^k/(2*k+1))^n, {k, 0, Infinity}] /. Pi -> 1 // Numerator; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Jun 20 2014 *)
Table[4/(2 Pi)^n LerchPhi[(-1)^n, n, 1/2], {n, 21}] // Numerator (* Eric W. Weisstein, Aug 02 2017 *)
Table[4/Pi^n If[Mod[n, 2] == 0, DirichletLambda, DirichletBeta][n], {n, 21}] // Numerator (* Eric W. Weisstein, Aug 02 2017 *)

{a(n) = if( n<0, 0, numerator( polcoeff( 1 / (1 - tan(x/4 + x * O(x^n))), n)))}; /* Michael Somos, Nov 11 2014 */

There is a simple formula in terms of Euler and Bernoulli numbers.
a(2n) = A046976(n), a(2n+1) = A089171(n+1) (conjectured).
Numerator of coefficients of expansion of (sec(x/2) + tan(x/2) + 1)/2 in powers of x. - Sergei N. Gladkovskii, Nov 11 2014

Entry revised by N. J. A. Sloane, Mar 24 2002

A245198 Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(-infinity, infinity).

Original entry on oeis.org

1, 0, 4, 0, 0, 4, 1, 9, 1, 1, 5, 2, 5, 9, 5, 2, 0, 5, 7, 2, 6, 5, 0, 2, 8, 4, 1, 2, 1, 7, 8, 9, 4, 2, 6, 9, 3, 1, 6, 8, 9, 0, 2, 6, 7, 0, 1, 8, 6, 6, 3, 1, 0, 5, 4, 8, 4, 8, 7, 9, 5, 5, 4, 0, 1, 0, 0, 0, 5, 3, 1, 5, 5, 6, 9, 8, 6, 3, 4, 3, 8, 6, 8, 0, 3, 0, 2, 8, 3, 1, 8, 3, 9, 5, 3, 7, 8, 7, 4, 3, 3, 6, 4, 3

Offset: 1

Jean-François Alcover, Jul 17 2014

The corresponding Landau-Kolmogorov inequality for the first and third derivative is ||f'|| <= C(3,1) ||f||^(2/3) ||f'''||^(1/3) [see S. Finch ref. for C(n,k) and the general derivative inequalities], where the real-valued function f is defined on (-infinity, infinity), the involved norm being the supremum norm, defined by ||f|| = sup |f(x)|.
Hadamard proved that if f is twice differentiable and both f and f'' are bounded, then ||f'|| <= sqrt(2) ||f||^(1/2) ||f''||^(1/2), and the constant C(2,1) = sqrt(2) is the best possible.
Kolmogorov determined best constants C(n,k), 1 <= k <= n, for the inequality between derivatives in terms of Favard constants (A050970/A050971). These formulas giving C(n,k) include special cases discovered by G. E. Shilov for small values of n and k.
[All comments made after Steven R. Finch].

			1.0400419115259520572650284121789426931689026701866310548487955401...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[3, 1], 10, 104] // First
(* or, directly: *) RealDigits[3^(2/3)/2, 10, 104] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(3,1) = 3^(2/3)/2 = (9/8)^(1/3).

A246006 a(2n) = numerator of |Bernoulli(2n)|, a(2n+1) = Euler(2n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 61, 1, 1385, 5, 50521, 691, 2702765, 7, 199360981, 3617, 19391512145, 43867, 2404879675441, 174611, 370371188237525, 854513, 69348874393137901, 236364091, 15514534163557086905, 8553103, 4087072509293123892361, 23749461029, 1252259641403629865468285

Offset: 0

Eric Chen, Nov 13 2014

nonn

Primes p which divide at least one a(n) for n<=p-2 are called weakly-irregular primes. For example, 19|a(11), 31|a(23), 37|a(32), 43|a(13), 47|a(15), 59|a(44), 61|a(7), ... - Eric Chen, Nov 26 2014
The weakly-irregular primes below 500 are 19, 31, 37, 43, 47, 59, 61, 67, 71, 79, 101, 103, 131, 137, 139, 149, 157, 193, 223, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 347, 349, 353, 359, 373, 379, 389, 401, 409, 419, 421, 433, 461, 463, 467, 491. - Eric Chen, Nov 26 2014
A prime can divide more than one a(n) for n<=p-2; for example, 67 divides both a(27) and a(58); additional examples are 101, 149, 157, 241, 263, 307, 311, ... . - Eric Chen, Nov 26 2014
Smallest values of k such that the n-th weakly-irregular prime divides a(k) are 11, 23, 32, 13, 15, 44, 7, 27, 29, 19, 63, 24, 22, 43, 129, 130, 62, 75, ... . - Eric Chen, Nov 26 2014
Smallest prime factors (>= n+2) of a(n) are 1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 19, 691, 43, 1, 47, 3617, 228135437, 43867, 79, 283, 41737, 131, 31, 103, 2137, 657931, 67, 9349, 71, ... . - Eric Chen, Nov 26 2014
The irregular pairs are (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ... . - Eric Chen, Nov 26 2014

			Euler(10) = 50521, so a(11) = 50521.
Bernoulli(12) = 691/2730, so a(12) = 691.

Eric Chen, Table of n, a(n) for n = 0..199
Peter Luschny, Irregular primes
Wikipedia, Irregular prime

Cf. A027641, A122045, A000367, A028296, A000111, A000364, A000182.

Cf. A050970, A050971, A068205.

a246006[n_] := If[EvenQ[n], Abs[Numerator[BernoulliB[n]]], Abs[EulerE[n-1]]]; Table[a246006[n], {n, 0, 99}]

from sympy import euler, bernoulli
def A246006(n): return abs(euler(n-1)) if n&1 else abs(bernoulli(n)).p # Chai Wah Wu, Apr 15 2023

A245294 Decimal expansion of the square root of 6/5.

Original entry on oeis.org

1, 0, 9, 5, 4, 4, 5, 1, 1, 5, 0, 1, 0, 3, 3, 2, 2, 2, 6, 9, 1, 3, 9, 3, 9, 5, 6, 5, 6, 0, 1, 6, 0, 4, 2, 6, 7, 9, 0, 5, 4, 8, 9, 3, 8, 9, 9, 9, 5, 9, 6, 6, 5, 0, 8, 4, 5, 3, 7, 8, 8, 8, 9, 9, 4, 6, 4, 9, 8, 6, 5, 5, 4, 2, 4, 5, 4, 4, 5, 4, 6, 7, 6, 0, 1, 7, 1, 6, 8, 7, 2, 3, 2, 7, 7, 4, 1, 2, 5, 1, 5, 2, 9, 4, 5

Offset: 1

Jean-François Alcover, Jul 17 2014

Decimal expansion of the Landau-Kolmogorov constant C(4,2) for derivatives in the case L_infinity(infinity, infinity).
See A245198.
Apart from the first digit the same as A176057. - R. J. Mathar, Jul 21 2014

			1.095445115010332226913939565601604267905489389995966508453788899464986554...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Landau-Kolmogorov Constants.
Eric Weisstein's MathWorld, Favard Constants.

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[4, 2], 10, 105] // First
RealDigits[Sqrt[6/5], 10, 100][[1]] (* Amiram Eldar, Jul 19 2022 *)

sqrt(6/5) \\ Charles R Greathouse IV, Aug 26 2017

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).
C(4,2) = sqrt(6/5).
Equals Sum_{k>=0} binomial(2*k,k)/24^k. - Amiram Eldar, Jul 19 2022

A162445 A sequence related to the Beta function.

Original entry on oeis.org

1, 8, 384, 46080, 2064384, 3715891200, 392398110720, 1428329123020800, 274239191619993600, 1678343852714360832000, 102043306245033138585600, 4714400748520531002654720000, 160144566965128191597871104000

Offset: 0

Johannes W. Meijer, Jul 06 2009

We define F(z) = Beta(1/2-z/2,1/2+z/2)/Beta(1/2,1/2) = 1/sin(Pi*(1+z)/2) with Beta(z,w) the Beta function. See A008956 for a closely related function.
For the Taylor series expansion of F(z) we can write F(z) = sum(b(n)*(Pi*z)^(2*n)/a(n), n=0..infinity) with b(n) = A046976(n) and a(n) the sequence given above.
We can also write F(z) = sum(c(n)*(Pi*z)^(2*n)/d(n), n=0..infinity) with c(n) = A000364(n) and d(n) = A067624(n).
If p(n) is the exponent of the prime factor 2 in a(n) than p(n) = A120738(n) and 2^p(n) = A061549(n) = abs((4*n)!!/A117972(n)).

Bisection of A050971
Equals 2^(2*n)*A046977(n)
Cf. A008956, A046976, A000364, A067624, A120738, A061549 and A117972.

Denominator[Table[EulerE[2n]/(4n)!!,{n,0,20}]] (* Harvey P. Dale, Jun 23 2013 *)

a(n) = denom(euler(2*n)/(4*n)!!)

A245293 Decimal expansion of the Landau-Kolmogorov constant C(4,1) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 0, 8, 0, 9, 6, 0, 1, 2, 3, 8, 4, 5, 6, 2, 7, 5, 1, 5, 1, 8, 8, 0, 8, 0, 1, 5, 0, 6, 3, 6, 5, 4, 5, 6, 4, 9, 2, 3, 7, 5, 7, 7, 0, 7, 4, 7, 2, 5, 5, 2, 3, 4, 3, 8, 0, 1, 3, 5, 6, 6, 4, 4, 2, 5, 9, 2, 7, 5, 9, 9, 0, 9, 7, 9, 0, 6, 6, 8, 5, 7, 2, 5, 0, 6, 8, 4, 8, 1, 8, 1, 1, 2, 7, 0, 7, 0, 7, 6, 1, 6, 1, 7, 7, 9

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.0809601238456275151880801506365456492375770747255234380135664425927599...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[4, 1], 10, 105] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(4,1) = 4*(2/3)^(1/4)/5^(3/4) = (512/375)^(1/4).

A245295 Decimal expansion of the Landau-Kolmogorov constant C(4,3) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 4, 8, 0, 1, 6, 5, 6, 0, 8, 9, 8, 4, 5, 7, 0, 5, 0, 1, 1, 3, 3, 5, 7, 9, 9, 3, 2, 3, 2, 7, 6, 7, 3, 6, 3, 8, 5, 9, 8, 1, 2, 3, 5, 8, 2, 6, 1, 2, 3, 7, 6, 2, 3, 6, 6, 4, 9, 7, 2, 4, 8, 1, 1, 8, 3, 1, 4, 9, 3, 3, 7, 3, 1, 5, 9, 9, 2, 3, 0, 5, 2, 4, 0, 8, 8, 8, 3, 9, 0, 3, 8, 0, 3, 7, 6, 7, 9, 7, 3, 4, 5, 1, 4, 9

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.480165608984570501133579932327673638598123582612376236649724811831493373...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[4, 3], 10, 105] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(4,3) = (3/5)^(1/4)*2^(3/4) = (24/5)^(1/4).

A245296 Decimal expansion of the Landau-Kolmogorov constant C(5,1) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 0, 4, 4, 2, 5, 7, 9, 0, 9, 3, 0, 9, 7, 9, 5, 1, 4, 3, 4, 4, 5, 3, 6, 9, 6, 1, 7, 1, 5, 5, 7, 0, 2, 5, 8, 3, 0, 8, 0, 4, 2, 0, 8, 0, 4, 2, 0, 2, 5, 3, 7, 2, 0, 7, 7, 5, 7, 6, 1, 3, 4, 1, 5, 8, 0, 0, 2, 3, 2, 5, 8, 8, 8, 0, 0, 6, 2, 3, 5, 7, 8, 8, 7, 4, 4, 6, 0, 2, 0, 1, 1, 1, 9, 2, 2, 0, 2, 7, 8, 5, 4, 7, 2, 4

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.0442579093097951434453696171557025830804208042025372077576134158002325888...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[5,1], 10, 105] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(5,1) = (5*5^(4/5))/(8*2^(4/5)*3^(1/5)) = (1953125/1572864)^(1/5).

A245297 Decimal expansion of the Landau-Kolmogorov constant C(5,2) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 1, 1, 6, 6, 4, 5, 9, 7, 1, 1, 0, 3, 8, 0, 9, 8, 8, 2, 6, 4, 5, 7, 1, 5, 4, 5, 1, 0, 7, 3, 1, 5, 3, 1, 7, 8, 9, 6, 6, 5, 1, 2, 0, 0, 6, 6, 9, 7, 4, 0, 4, 0, 1, 6, 4, 5, 6, 3, 4, 2, 1, 6, 0, 6, 0, 8, 1, 7, 9, 5, 2, 8, 6, 4, 8, 5, 2, 2, 2, 9, 6, 8, 4, 6, 4, 6, 0, 0, 2, 6, 2, 2, 4, 5, 4, 9, 9, 1, 2, 3

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.1166459711038098826457154510731531789665120066974040164563421606...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[5,2], 10, 101] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(5,2) = (5*5^(4/5))/(8*2^(4/5)*3^(1/5)) = (1953125/1572864)^(1/5).

A245298 Decimal expansion of the Landau-Kolmogorov constant C(5,3) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 1, 1, 9, 4, 2, 3, 7, 3, 1, 7, 3, 5, 1, 0, 7, 6, 1, 1, 6, 2, 9, 7, 1, 1, 0, 8, 2, 0, 8, 1, 2, 6, 1, 0, 4, 1, 2, 4, 9, 9, 8, 5, 5, 6, 7, 0, 5, 8, 6, 0, 7, 0, 8, 6, 5, 2, 0, 9, 8, 2, 7, 9, 9, 1, 3, 1, 5, 4, 2, 2, 9, 2, 2, 9, 6, 9, 0, 4, 5, 1, 5, 2, 5, 2, 6, 2, 8, 6, 5, 9, 6, 1, 3, 0, 8, 5, 2, 2, 9, 2, 9, 5, 2

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.11942373173510761162971108208126104124998556705860708652098279913...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[5,3], 10, 104] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(5,3) = (1/2)*(15/2)^(2/5).

cp's OEIS Frontend

A050970 Numerator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} (4k+1)^(-n).

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A245198 Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(-infinity, infinity).

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A246006 a(2n) = numerator of |Bernoulli(2n)|, a(2n+1) = Euler(2n).

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A245294 Decimal expansion of the square root of 6/5.

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A162445 A sequence related to the Beta function.

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A245293 Decimal expansion of the Landau-Kolmogorov constant C(4,1) for derivatives in the case L_infinity(infinity, infinity).

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A245295 Decimal expansion of the Landau-Kolmogorov constant C(4,3) for derivatives in the case L_infinity(infinity, infinity).

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