A228115 -id:A228115 - cp's OEIS frontend

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

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

Vladimir Pletser, Aug 10 2013

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.

Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
Eric Weisstein's World of Mathematics, Prime Counting Function.

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

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

A228111 Integer nearest to (S(n)F(4n)(S(n))), where F(4n)(x) are Fibonacci polynomials of multiple of 4 indices (4n) 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, 8385, 68929, 584467, 5074924, 44885325, 402777151, 3656032622, 33492393634, 309106153431, 2870123507479, 26783122426197, 250971797533095, 2359952229466124, 22256979400698116, 210440626023838163, 1994088284872617955, 18931694933036811169

Offset: 1

Vladimir Pletser, Aug 10 2013

nonn

Coefficients are A=16103485019141/2900449771918928, B=5262046568827901/29305205016290, C(0)=296261685121849/265642652464758, C(1)=38398556529727/750568780742436, C(2)=0, C(3)=-11594434149768/8254020049890781.
This sequence gives a good approximation of the number of primes with n digits (A006879); see (A228112).
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.  However, when using Fibonacci polynomials, the exact relation is xF(4n)(x)=(F(2n+1)(x))^2- (F(2n-1)(x))^2.
To obtain this sequence, one switches to the product of x and multiple of 4 index Fibonacci polynomials F(4n)(x), and one obtains the sequence a(n) by computing x as a function of n such that (xF(4n)(x)) fit the values of pi(10^n)- pi(10^(n-1)) for 1 <= n <= 25, with pi(1)=0.

			For n=1, xF(4)(x) = x^2*(x^2+2); replace x with Sum_{i=0..3} (C(i)*(log(log(A*(B+1))))^i)= 1.11173... to obtain a(1) = round((1.11173...)*F(4)(1.11173...)) = 4.
For n=2, xF(8)(x) = x^2*(x^2+2)*(x^4+4*x^2+2); replace x with Sum_{i=0..3} (C(i)*(log(log(A*(B+4))))^i)= 0.99998788... to obtain a(2) = round((0.99998788...)*F(8)(0.99998788...)) = 21.

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

Vladimir Pletser, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.

Cf. A006879, A228112, A228113, A228114, A228115, A228116.

with(combinat):A:=16103485019141/2900449771918928: B:=5262046568827901/29305205016290: C(0):=296261685121849/265642652464758: C(1):=38398556529727/750568780742436: C(2):=0: C(3):=-11594434149768/8254020049890781: b:=n->log(log(A*(B+n^2))): c:=n->sum(C(i)*(b(n))^i, i=0..3): seq(round(c(n)*fibonacci(4*n, c(n))), n=1..25);

a(n) = round(S(n)*F(4n)(S(n))), where S(n) = Sum_{i=0..3} (C(i)*(log(log(A*(B+n^2))))^i).

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

Vladimir Pletser, Aug 10 2013

sign

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.

Eric Weisstein's World of Mathematics, Prime Counting Function.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.

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

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

Vladimir Pletser, Aug 10 2013

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.

Eric Weisstein's World of Mathematics, Prime Counting Function.
Eric Weisstein's World of Mathematics, Riemann Prime Counting Function.

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

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

Vladimir Pletser, Aug 10 2013

nonn

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.

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

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

Vladimir Pletser, Table of n, a(n) for n = 1..100 [a(40)-a(41) corrected by Sean A. Irvine]
Eric Weisstein's World of Mathematics, Prime Counting Function.
Eric Weisstein's World of Mathematics, Riemann Prime Counting Function.

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

a(n) = A057793(n) - A057793(n-1).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A228111 Integer nearest to (S(n)*F(4n)(S(n))), where F(4n)(x) are Fibonacci polynomials of multiple of 4 indices (4n) and S(n) = Sum_{i=0..3} (C(i)*(log(log(A*(B+n^2))))^i) (see coefficients A, B, C(i) in comments).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A228113 First differences of A057793.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A228111 Integer nearest to (S(n)F(4n)(S(n))), where F(4n)(x) are Fibonacci polynomials of multiple of 4 indices (4n) and S(n) = Sum_{i=0..3} (C(i)(log(log(A*(B+n^2))))^i) (see coefficients A, B, C(i) in comments).