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 1-3 of 3 results.

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

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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).

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

Views

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).

A228063 Integer nearest to F[4n](S(n)), where F[4n](x) are Fibonacci polynomials and S(n) = Sum_{i=0..3} (C(i)*(log(log(A*(B+n^2))))^i) (see coefficients A, B, C(i) in comments).

Original entry on oeis.org

4, 21, 143, 1063, 8371, 68785, 583436, 5069633, 44876757, 403025174, 3660702622, 33550877248, 309726969451, 2876065468123, 26835315229835, 251389798269317, 2362887262236150, 22272676889496853, 210455460654786509, 1992806263723883464
Offset: 1

Views

Author

Vladimir Pletser, Aug 06 2013

Keywords

Comments

Coefficients are A=6.74100517717340111e-03, B=147.60482223254, C(0)=1.112640536670862472, C(1)=5.2280866355335360415e-02, C(2)=0, C(3)=-1.5569578292261924e-03.
This sequence gives a good approximation of the number of primes with n digits (A006879); see A228064.
As the squares of odd-indexed Fibonacci numbers F[2n+1](1) (see A227693) are equal or close to the first values of pi(10^n) (A006880), and as F[4n](1)=(F[2n+1](1))^2- (F[2n-1](1))^2, it is legitimate to ask whether the first values of the differences pi(10^n)- pi(10^(n-1)) (A006879) are also close or equal to multiple of 4 index Fibonacci numbers F[4n](1); e.g., for n=2, F[8](1)=21.
To obtain this sequence, one switches to multiple of 4 index Fibonacci polynomials F[4n](x), one obtains the sequence a(n) by computing x as a function of n such that F[4n](x) fit the values of pi(10^n)- pi(10^(n-1)) for 1 <= n <= 25, with pi(1)=0.

Examples

			For n =1, F[4](x) = x^3+2x; replace x by Sum_{i=0..3} (C(i)*(log(log(A*(B+1))))^i)= 1.179499… to obtain a(1)= round(F[4]( 1.179499...))=4. For n=2, F[8](x) = x^7+6x^5+10x^3+4x; replace x by Sum_{i=0..3} (C(i)*(log(log(A*(B+4))))^i)= 0.999861... to obtain a(2)= round(F[8]( 0.999861…))=21

References

  • Jonathan Borwein, David H. Bailey, Mathematics by Experiment, A. K. Peters, 2004, p. 65 (Table 2.2).
  • 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, A228064, A227693.

Programs

  • Maple
    with(combinat):A:=6.74100517717340111e-03: B:=147.60482223254: C(0):=1.112640536670862472: C(1):=5.2280866355335360415e-02: C(2):=0: C(3):=-1.5569578292261924e-03: b:=n->log(log(A*(B+n^2))): c:=n->sum(C(i)*(b(n))^i, i=0..3): seq(round(fibonacci(4*n, c(n))), n=1..25);

Formula

a(n) = round(F[4n](Sum_{i=0..3} (C(i)*(log(log(A*(B+n^2))))^i)) ).
Showing 1-3 of 3 results.

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