cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A228112 Difference between the number of primes with n digits (A006879) and the 6-parametric approximation of that number in A228111.

Original entry on oeis.org

0, 0, 0, -2, -22, -23, 1614, 21952, 200754, 1427826, 6969680, -2536429, -648528610, -11247293516, -143493754330, -1578026921839, -15633412845816, -140582270611489, -1122913035234416, -7326349588043722, -25245049578998081, 301375487087871682, 9140885960557495580, 157255672291012140238, 2265259467069624459434
Offset: 1

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Author

Vladimir Pletser, Aug 10 2013

Keywords

Comments

A228111 provides exact values of pi(10^n) - pi(10^(n-1)) for n = 1 to 3 and yields an average relative difference in absolute value, i.e. average(abs(A228112(n))/A006879(n) = 0.00375341... for 1 <= n <= 25, better than when using the 10^n/log(10^n) function, which yields 0.0469094... (see A228066) or the logarithmic integral (Li(10^n) - Li(2)) function, which yields 0.0175492... (see A228068) or the Riemann (Riemann(10^n)) function, which yields 0.0103936... (see A228114) or the Fibonacci polynomials of multiple of 4 indices, which yields 0.00473860... (see A228064) for 1 <= n <= 25.

Links

Crossrefs

Cf. A006880, A006879, A228063, A228066, A228068, A228111, A228113, A228114, A228115, A228116.

Formula

a(n) = A006879(n)- A228111(n).

A228115 First differences of A227693.

Original entry on oeis.org

3, 21, 143, 1061, 8366, 68932, 585881, 5094722, 45074595, 404185377, 3663479699, 33498077106, 308548877876, 2859703657128, 26646019345842, 249434445759050, 2344494354096166, 22116172789221197, 209301155352811190, 1986521422431963549, 18904049485198437478, 180323870540071281301, 1723847795281971132487, 16512536418951055856540, 158463448213030472998711
Offset: 1

Views

Author

Vladimir Pletser, Aug 10 2013

Keywords

Comments

This sequence gives a good approximation of the number of primes with n digits (A006879); see (A228116).

Examples

			For n =1, A227693(1)- A227693(0) =4-1=3, where A227693(1)= round((F[3]( 1.016825…))^2)=4 with F[3](x) = x^2+1 and A227693(0)= round(F[1](x)) =1 with F[1](x)=1

References

  • John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1996, page 144.

Links

Crossrefs

Cf. A006879, A228111 - A228116

Formula

a(n)= A227693(n)- A227693(n-1)

A228116 a(n) = A006879(n) - A228115(n).

Original entry on oeis.org

1, 0, 0, 0, -3, -26, 200, 2154, 11484, 19600, -477397, -8219901, -91253055, -827443165, -6390673975, -40675147794, -175537475858, 224340865430, 16557635792557, 240512852610684, 2400398259375610, 16146663225893061, 5309635516930146, -2257043208658957597, -52738581235904454897
Offset: 1

Views

Author

Vladimir Pletser, Aug 10 2013

Keywords

Comments

Difference between the number of primes with n digits (A006879) and its estimate by squares of odd-indexed Fibonacci polynomials (A228115).
The sequence (A228115) provides exactly the values of pi(10^n)- pi(10^(n-1)) for n=2 to 4 and yields an average relative difference in absolute value, i.e. Average(Abs(A228116(n))/ (A006879(n)) = 1.01656…x10^-2 for 1<=n<=25, better than when using the ((10^n)/log(10^n)) function (Average(Abs(A228066(n))/ (A006879(n)) = 4.69094…x10^-2 (see A228066)), or the Logarithm integral (Li(10^n)-Li(2)) function (Average(Abs(A228068(n))/ (A006879(n)) = 1.75492…x10^-2 (see A228068)), or the Riemann(Riemann (10^n)) function (Average(Abs(A228114(n))/ (A006879(n)) = 1.03936…x10^-2) for 1<=n<=25.
Furthermore, if the first value for n=1 is skipped, the average relative difference in absolute value is improved by nearly two orders of magnitude, i.e. Average(Abs(A228116(n))/ (A006879(n)) = 1.72564…x10^-4 for 2<=n<=25, better than when using the ((10^n)/log(10^n)) function (Average(Abs(A228066(n))/ (A006879(n)) = 4.88640…x10^-2 (see A228066)), or the Logarithm integral (Li(10^n)-Li(2)) function (Average(Abs(A228068(n))/ (A006879(n)) = 7.86383…x10^-3 (see A228068)), or the Riemann(Riemann (10^n)) function (Average(Abs(A228114(n))/ (A006879(n)) = 4.10042…x10^-4), or the product of x and Fibonacci polynomials of multiple of 4 indices F[4n](x) (Average(Abs(A228064(n))/ (A006879(n)) = 3.90981…x10^-3 (see A228112)) for 2<=n<=25.

Links

Crossrefs

Cf. A006880, A006879, A228063, A228066, A228068, A228111-A228115.

Formula

a(n) = A006879(n) - A228115(n).

A228114 Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to Riemann(10^n) (see A228113).

Original entry on oeis.org

-1, 0, 1, 2, 3, -34, -59, -9, 176, 1749, 490, -842, 4297, 13427, -92418, -253834, 925307, 2903111, -27385699, 28776158, 81540379, 40700461, -1160432518, 2692289572, 175794995
Offset: 1

Views

Author

Vladimir Pletser, Aug 10 2013

Keywords

Comments

The sequence (A228113) yields an average relative difference in absolute value, i.e. Average(Abs(A228114(n))/ (A006879(n)) = 1.03936…x10^-2 for 1<=n<=25.
Note that A057793(n) = Riemann(10^n) is not defined for n=0. Its value is set to 0.

Links

Crossrefs

Cf. A006880, A006879, A057793, A228111, A228112, A228113, A228115, A228116.

Formula

a(n) = A006879(n) - A228113(n).

A228113 First differences of A057793.

Original entry on oeis.org

5, 21, 142, 1059, 8360, 68940, 586140, 5096885, 45085903, 404203228, 3663001812, 33489858047, 308457620524, 2858876200536, 26639628764285, 249393770865090, 2344318815695001, 22116397127183516, 209317713015989446, 1986761935255798075, 18906449883376272709
Offset: 1

Views

Author

Vladimir Pletser, Aug 10 2013

Keywords

Comments

This sequence is an approximation to the number of primes with n digits (A006879). The error in the approximation is tabulated in A228114.
Because A057793(n) = Riemann(10^n) is not defined for n=0, we set its value to zero for our purpose of defining the differences.

Examples

			For n=1, A057793(1) - A057793(0) = 5 - 0 = 5.

References

  • John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, page 144-146.

Links

Crossrefs

Cf. A006879, A057793, A228111, A228112, A228114, A228115, A228116.

Formula

a(n) = A057793(n) - A057793(n-1).
Showing 1-5 of 5 results.

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