A070769 -id:A070769 - cp's OEIS frontend

A099805 Positions of records for terms in the continued fraction of Soldner's constant (A070769).

1, 2, 3, 12, 70, 126, 202, 585, 1592, 2436, 2544, 4814, 9603, 12148, 122447

Offset: 1

Eric W. Weisstein, Oct 26 2004

See A229230 for another version.

Eric Weisstein's World of Mathematics, Soldner's Constant

Cf. A229230 (= a(n) - 1), A070769, A099804.

f = FindRoot[LogIntegral@x, {x, 3/2}, WorkingPrecision -> 2^17][[1, 2]]; cf = ContinuedFraction@f; k = 1; mx = 0; lst = {}; len = Length@ cf; While[k < len, If[ cf[[k]] > mx, mx = cf[[k]]; AppendTo[lst, k]]; k++ ]; lst (* Robert G. Wilson v, Aug 05 2010 *)

a(15) from Robert G. Wilson v, Aug 05 2010

Edited by N. J. A. Sloane, Jun 19 2021 (removed mention of a(0) from definition).

A380270 Decimal expansion of Integral_{x=1..A070769} li(x) dx (negated), where li(x) is the logarithmic integral.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 18 2025

A070769 is Soldner's constant, where li(A070769)=0.

Integral_{x=0..1} li(x) dx = -log(2) then Integral_{x=0..A070769} li(x) dx = A380270 - log(2) = -1.19324951683011591582919049...

			-0.500102336270170606411958373..

Cf. A069284, A070769, A084945, A091723, A244067, A257817, A257818, A257819, A257820, A257821, A270857.

y = x /. FindRoot[LogIntegral[x] == 0, {x, 1.5}, WorkingPrecision -> 200]; yy = -Integrate[LogIntegral[x], {x, 1, y}]; RealDigits[yy, 10, 105][[1]]

A227539 Signature sequence of Soldner's constant (A070769).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 8, 1, 4, 7, 3, 6, 9, 2, 5, 8, 1, 4, 7, 10, 3, 6, 9, 2, 5, 8, 11, 1, 4, 7, 10, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 13, 3, 6, 9, 12, 2, 5, 8, 11, 14, 1, 4, 7, 10, 13, 3, 6, 9, 12, 15, 2, 5, 8

Offset: 1

Casey Mongoven, Jul 16 2013

nonn

Arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x; the sequence of j's is the signature of 1/x.
The plot looks surprisingly regular. - T. D. Noe, Jul 23 2013
Where does this first differ from A133334? - R. J. Mathar, Jul 30 2013

Clark Kimberling, Fractal Sequences and Interspersions, Ars Combinatoria 45 (1997) 157-168.

Eric Weisstein's World of Mathematics, Soldner's Constant.
Eric Weisstein's World of Mathematics, Logarithmic Integral.
Index entries for sequences related to signature sequences

Cf. A070769, A099803.

x = FindRoot[LogIntegral[x] == 0, {x, 2}, WorkingPrecision -> 105][[1,2]]; Take[Transpose[Sort[Flatten[Table[{i + j*x, i}, {i, 30}, {j, 20}], 1], #1[[1]] < #2[[1]] &]][[2]], 100] (* T. D. Noe, Jul 23 2013 *)

A091723 Decimal expansion of the root x of Ei(x)=0, where Ei is the exponential integral.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Feb 01 2004

			0.372507410781366634461991866580119133535689497771654...

Robert Price, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Exponential Integral.
Wikipedia, Exponential integral.

Cf. A070769, A084945.

RealDigits[ x /. FindRoot[ ExpIntegralEi[x] == 0, {x, 1}, WorkingPrecision -> 102]][[1]] (* Jean-François Alcover, Oct 29 2012 *)
RealDigits[x /. FindRoot[LogIntegral[Exp[x]]/x, {x, 1/3}, WorkingPrecision -> 105]][[1]] (* Artur Jasinski, Apr 19 2022 *)

solve(x=.3,1,real(eint1(-x))) \\ Charles R Greathouse IV, Apr 14 2014

Equals log(A070769). - Amiram Eldar, Aug 14 2020

Equals root x of li(exp(x)/x)=0 where li(x) is the logarithmic integral. - Artur Jasinski, Apr 19 2022

A257821 Decimal expansion of the unique real number a>0 such that the real part of li(-a) is zero.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, May 11 2015

As discussed in A257819, the real part of li(z) is a well behaved function for any real z, except for the singularity at z=+1. It has three roots: z=A070769 (the Soldner's constant), z=0, and z=-a. However, unlike in the other two cases, the imaginary part of li(-a) is not infinitesimal in the neighborhood of this root; it is described in A257822.

			2.4664082624126780751971033507759329502907808782774099823786...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Logarithmic integral function

Cf. A070769, A257819, A257822.

RealDigits[a/.FindRoot[Re[LogIntegral[-a]]==0,{a,2},WorkingPrecision->120]][[1]] (* Vaclav Kotesovec, May 11 2015 *)

li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
  if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
  return(+Pi*I-eint1(-log(c)))); }
  a=-solve(x=-3,-1,real(li(x)))  \\ Better use excess realprecision

Satisfies real(li(-a)) = 0.

A099803 Continued fraction for Soldner's constant.

Original entry on oeis.org

1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, 7, 2, 1, 1, 1, 2, 30, 6, 3, 6, 1, 6, 1, 3, 1, 1, 1, 1, 10, 1, 1, 2, 2, 1, 4, 2, 5, 26, 5, 1, 19, 2, 6, 1, 10, 28, 1, 2, 1, 10, 2, 1, 1, 11, 1, 4, 19, 16, 99, 1, 1, 1, 35, 1, 1, 3, 2, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 11, 3, 2, 1, 6, 5, 1

Offset: 0

Eric W. Weisstein, Oct 26 2004

			[1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, ...]

Eric Weisstein's World of Mathematics, Soldner's Constant.

Cf. A070769 (decimal expansion), A099804 (records).

ContinuedFraction[x/.FindRoot[LogIntegral[x]==0,{x,1.4}, WorkingPrecision-> 120]] (* Harvey P. Dale, May 01 2012 *)

Offset changed by Andrew Howroyd, Aug 04 2024

A099804 Records for terms in the continued fraction of Soldner's constant.

Original entry on oeis.org

1, 2, 4, 47, 99, 294, 527, 616, 1152, 1456, 2638, 2705, 4105, 31772, 88683, 90658, 95845, 237245, 1387437, 5499408, 12672729, 110482114

Offset: 1

Eric W. Weisstein, Oct 26 2004

Eric Weisstein's World of Mathematics, Soldner's Constant Continued Fraction

Cf. A070769 (decimal expansion of Soldner's constant).

Cf. A099805, A229230.

f = FindRoot[LogIntegral@x, {x, 3/2}, WorkingPrecision -> 2^17][[1, 2]]; cf = ContinuedFraction@f; k = 1; mx = 0; lst = {}; len = Length@ cf; While[k < len, If[cf[[k]] > mx, mx = cf[[k]]; AppendTo[lst, cf[[k]]]]; k++ ]; lst (* Robert G. Wilson v, Aug 05 2010 *)

a(15) from Robert G. Wilson v, Aug 05 2010
More terms from Eric W. Weisstein, Sep 16 2013
a(21)-a(22) from Eric W. Weisstein, Oct 07 2013

A122421 Soldner primes: primes formed from concatenation of decimal digits of Soldner's constant.

Original entry on oeis.org

1451, 145136923488338105028396848589202744949303228364801586309300455766242559575451783565953135771108682884704075157097064920307143357020423478488319

Offset: 1

Eric W. Weisstein, Sep 03 2006

Amiram Eldar, Table of n, a(n) for n = 1..4
Eric Weisstein's World of Mathematics, Constant Primes.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Cf. A070769, A122422.

A122422 Numbers n such that first n digits of Soldner's constant gives a prime number.

Original entry on oeis.org

4, 144, 227, 444, 19474

Offset: 1

Eric W. Weisstein, Sep 03 2006

Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Soldner's Constant

Cf. A070769, A122421.

a(5) = 19474 from Eric W. Weisstein, Sep 06 2006

A229230 Positions of incrementally largest terms in the continued fraction of Soldner's constant.

Original entry on oeis.org

0, 1, 2, 11, 69, 125, 201, 584, 1591, 2435, 2543, 4813, 9602, 12147, 122446, 172037, 185124, 203717, 319849, 471163, 3876852, 4491253

Offset: 1

Eric W. Weisstein, Sep 16 2013

Correctly indexed version of A099805 with [a_0; a_1, a_2, ...] = [1; 2, 4, 1, 1, 1, 3, ...]

Eric Weisstein's World of Mathematics, Soldner's Constant Continued Fraction

Cf. A070769 (decimal expansion of Soldner's constant).

Cf. A099805 (= a(n) - 1).

a(n) = A099805(n) - 1.

a(21)-a(22) from Eric W. Weisstein, Oct 07 2013

cp's OEIS Frontend

A099805 Positions of records for terms in the continued fraction of Soldner's constant (A070769).

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A380270 Decimal expansion of Integral_{x=1..A070769} li(x) dx (negated), where li(x) is the logarithmic integral.

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A227539 Signature sequence of Soldner's constant (A070769).

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A091723 Decimal expansion of the root x of Ei(x)=0, where Ei is the exponential integral.

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A257821 Decimal expansion of the unique real number a>0 such that the real part of li(-a) is zero.

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A099803 Continued fraction for Soldner's constant.

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A099804 Records for terms in the continued fraction of Soldner's constant.

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Programs

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Extensions

A122421 Soldner primes: primes formed from concatenation of decimal digits of Soldner's constant.

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A122422 Numbers n such that first n digits of Soldner's constant gives a prime number.

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