A070769 -id:A070769 - cp's OEIS frontend

A276762 Decimal expansion of the real part of a fixed point of the logarithmic integral li(z) in C.

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

Offset: 1

Stanislav Sykora, Oct 28 2016

The function li(z) has in C two fixed points, namely z0 = A276762 + A276763*i and its conjugate. For real arguments, z = 0 is also a fixed point but, since it lies on a branch cut and has discontinuous imaginary part, setting li(0) = 0 in C is problematic. This leaves z_0 and its conjugate as the only 'regular' fixed points of li(z). They are both attractors of the mapping. The attraction basin of z0 appears to be the whole upper half of the complex plane (including the real axis, minus the singular points z = 0 and z =1), while that of the conjugate of z0 appears to be the whole lower half (exluding the real axis). The convergence is exponential but rather slow, with the asymptotic convergence factor of 0.756330... per iteration, equal to abs(li'(z0)) = 1/abs(log(z0)).

			1.878881747908123091969486497326883469689594952855636606894527515...

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

Cf. A276763 (imaginary part), A070769.

RealDigits[Re[z/.FindRoot[LogIntegral[z] == z, {z, 2+I}, WorkingPrecision -> 100]]][[1]] (* Vaclav Kotesovec, Oct 30 2016 *)

\\ z may be t_INT, t_REAL, or t_COMPLEX except 0 or 1
li(z)=
{
  my(sgn=(-1)^if(real(z)<1,imag(z)<0,imag(z)<=0));
  sgn*Pi*I - eint1(-log(z));
}
default(realprecision,2100); \\ Execution:
Eps_= 4.0*10.0^(-default(realprecision));
z=1+I;zlast=0; \\ Initialize and iterate
for(k=1,1e6,z=li(z);if(abs(z-zlast)

Setting z0 = A276762 + A276763*i, li(z0) = z0.

A276763 Decimal expansion of the imaginary part of a fixed point of the logarithmic integral li(z) in C.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Oct 28 2016

See A276762 for the real part, as well as detailed comments and links.

			2.06592220237066218898810461125410843001424985319006732838579118...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A276762 (real part), A070769.

RealDigits[Im[z/.FindRoot[LogIntegral[z] == z, {z, 2+I}, WorkingPrecision -> 100]]][[1]] (* Vaclav Kotesovec, Oct 30 2016 *)

\\ z may be t_INT, t_REAL, or t_COMPLEX except 0 or 1
li(z)=
{
  my(sgn=(-1)^if(real(z)<1,imag(z)<0,imag(z)<=0));
  sgn*Pi*I - eint1(-log(z));
}
default(realprecision,2100); \\ Execution:
Eps_= 4.0*10.0^(-default(realprecision));
z=1+I;zlast=0; \\ Initialize and iterate
for(k=1,1e6,z=li(z);if(abs(z-zlast)

A355080 Start with the positive integers. Term by term from left to right insert a copy of the current term x m steps further, where m is the number of times x has appeared.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 3, 1, 3, 2, 4, 4, 3, 4, 1, 2, 4, 3, 5, 5, 1, 5, 4, 2, 5, 3, 6, 6, 4, 6, 5, 1, 6, 2, 3, 5, 6, 4, 7, 7, 1, 7, 2, 6, 7, 5, 3, 4, 7, 8, 8, 6, 8, 1, 5, 8, 7, 2, 3, 8, 6, 4, 7, 9, 9, 8, 9, 5, 1, 9, 2, 6, 8, 9, 7, 3, 4, 5, 9, 10, 10, 8, 10, 1, 7, 10, 6

Offset: 1

Thomas Scheuerle, Jun 18 2022

We start with the sequence of positive integers 1, 2, 3, ... . We process number by number from left to right. If the number is found the first time in the sequence, we will place a copy of it directly after, by shifting the remaining part of the sequence to the right. If we have already seen this number m times, we will place a copy of this number after skipping m numbers on the right.
We could use the term ordinal instead of number here, as the numerical value itself is not important, only the ordering.
This sequence was inspired by sequence A354223 from Tamas Sandor Nagy, which shares the idea to start with a predefined sequence and to insert copies ahead of element-wise evaluation.
A mysterious constant C:
The indices where this sequence reaches the next greater number for the first time are roughly approximated by a parabola: a(floor(b+(1/C)*n^2)) = 1, 2, 3, ... .
a(k) approximates round(sqrt(C*k)) if we choose for k the indices where a new number appears the first time in this sequence.
For each number in this sequence the indices of the appearance can be roughly approximated by some polynomial b+(1/C)*n^2, where b is some individual constant for each number, but C always appears to be the same constant, known thus far to be 1.1738... . The author used the value of sqrt(2/u), where u is Soldner's constant, with very good results, but there is yet not any evidence known that Soldner's constant has any relation to this sequence. Can we estimate C more accurately? Can we find an expression or series to describe C?
Tamas Sandor Nagy noticed that the value 1 appears exactly once between the first appearances of any two consecutive record values. He further noticed that if we break this sequence up into an irregular triangle in which each record value starts a new row, we will observe columns (A000124) which show a progression with the row number. See example section for details.
The mean value of the rows mentioned above as a function of the row index r is approximately r/(Pi*log(2)^2) - 1/2.

			Step-by-step development of the sequence is as follows; the asterisk marks the actual term that will be processed:
* 1 was previously seen 0 times -> insert directly after.
1,2,3,4,5,6,7,8,9,10
1,1,2,3,4,5,6,7,8,9,10
  * 1 was previously seen once -> insert one later.
1,1,2,3,4,5,6,7,8,9,10
1,1,2,1,3,4,5,6,7,8,9,10
    * 2 was previously seen 0 times -> insert directly after.
1,1,2,1,3,4,5,6,7,8,9,10
1,1,2,2,1,3,4,5,6,7,8,9,10
      * 2 was previously seen once -> insert one later.
1,1,2,2,1,3,4,5,6,7,8,9,10
1,1,2,2,1,2,3,4,5,6,7,8,9,10
        * 1 was previously seen twice -> insert two later.
1,1,2,2,1,2,3,4,5,6,7,8,9,10
1,1,2,2,1,2,3,1,4,5,6,7,8,9,10
.
This sequence written as an irregular triangle:
*  *
1, 1     *
2, 2, 1, 2
3, 3, 1, 3, 2     *
4, 4, 3, 4, 1, 2, 4, 3
5, 5, 1, 5, 4, 2, 5, 3        *
6, 6, 4, 6, 5, 1, 6, 2, 3, 5, 6, 4
Each column below an asterisk shows a linear progression.

Thomas Scheuerle, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, Scatter plot of the first 10000 values
Thomas Scheuerle, y = k - (1/C)*(x-0.283)^2, where k is the least k such that a(k) = x. For C we chose sqrt(2/u) with u = 1.45... (A070769).
Thomas Scheuerle, y = k - (1/C)*(x-1.275)^2, where k is such that a(k) = 1 for the x-th appearance. For C we chose sqrt(2/u) with u = 1.45... (A070769).

Cf. A000124, A354223.

function a = A355080( max_n )
    a = 1:max_n;
    for n = 1:max_n
        j = length(find(a(1:n) == a(n)));
        a = [a(1:n+j-1) a(n) a(n+j:end)];
    end
    a = a(1:max_n);
end

A229201 Beginning position of n in the decimal expansion of Soldner's constant.

Original entry on oeis.org

17, 1, 8, 5, 2, 3, 6, 34, 11, 7, 16, 93, 258, 4, 1, 51, 241, 78, 284, 143, 31, 240, 43, 8, 67, 69, 259, 33, 20, 216, 40, 87, 42, 13, 9, 81, 5, 174, 14, 22, 105, 191, 68, 125, 35

Offset: 0

Eric W. Weisstein, Sep 15 2013

Eric W. Weisstein, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Constant Digit Scanning
Eric Weisstein's World of Mathematics, Soldner's Constant Digits

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

A276709 Decimal expansion of the derivative of logarithmic integral at its positive real root.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Sep 15 2016

Since the real root location of li(x) is the Soldner's constant A070769, this constant equals 1/log(A070769). It is also the inverse of the unique real root A091723 of the exponential integral function Ei(x).

			2.68451035082070765250238264048723868531017973459855163522041486450...

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, A091723, A257821.

1/x/.FindRoot[ExpIntegralEi[x] == 0, {x, 1}, WorkingPrecision -> 104] (* Vaclav Kotesovec, Sep 27 2016 *)

li(z) = {my(c=z+0.0*I); \\ Computes li(z) for any complex z
if(imag(c)<0,return(-Pi*I-eint1(-log(c))),return(+Pi*I-eint1(-log(c))));}
a = 1/log(solve(x=1.1,2.0,real(li(x)))) \\ Computes this constant

Equals 1/log(A070769) and 1/A091723.

cp's OEIS Frontend

A276762 Decimal expansion of the real part of a fixed point of the logarithmic integral li(z) in C.

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A276763 Decimal expansion of the imaginary part of a fixed point of the logarithmic integral li(z) in C.

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A355080 Start with the positive integers. Term by term from left to right insert a copy of the current term x m steps further, where m is the number of times x has appeared.

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A229201 Beginning position of n in the decimal expansion of Soldner's constant.

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A276709 Decimal expansion of the derivative of logarithmic integral at its positive real root.

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Mathematica

PARI

Formula

A229071 First occurrence of n consecutive n's in the decimal expansion of Soldner's constant.

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