A227693 -id:A227693 - cp's OEIS frontend

A228115 First differences of A227693.

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

Vladimir Pletser, Aug 10 2013

nonn

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

			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

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, A228111 - A228116

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

A227694 Difference between pi(10^n) and nearest integer to (F[2n+1](S(n)))^2 where pi(10^n) = number of primes <= 10^n (A006880), F[2n+1](x) are Fibonacci polynomials of odd indices [2n+1] and S(n) = Sum_{i=0..2} (C(i)(log(log(A(B+n^2))))^(2i)) (see A227693).

Original entry on oeis.org

0, 0, 0, 0, -3, -29, 171, 2325, 13809, 33409, -443988, -8663889, -99916944, -927360109, -7318034084, -47993181878, -223530657736, 810207694, 16558446000251, 257071298610935, 2657469557986545, 18804132783879606, 24113768300809752, -2232929440358147845, -54971510676262602742

Offset: 1

Vladimir Pletser, Jul 19 2013

sign

A227693 provides exactly the values of pi(10^n) for n = 1 to 4 and yields an average relative difference in absolute value, average(abs(A227694(n))/pi(10^n)) = 1.58269...*10^-4 for 1 <= n <= 25.

A227693 provides a better approximation to the distribution of pi(10^n) than: (1) the Riemann function R(10^n) as the sequence of integers nearest to R(10^n) (A057794), which yields 0.01219...; (2) the functions of the logarithmic integral Li(x) = Integral_{t=0..x} dt/log(t), whether as the sequence of integers nearest to (Li(10^n) - Li(3)) (A223166), which yields 0.0074969... (see A223167), or as Gauss's approximation to pi(10^n), i.e., the sequence of integers nearest to (Li(10^n) - Li(2)) (A190802), which yields 0.020116... (see A106313), or as the sequence of integer nearest to Li(10^n) (A057752), which yields 0.032486....

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.

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

Cf. A006880, A225137, A215663, A057794, A223166, A223167, A190802, A106313, A057752, A227693.

a(n) = A006880(n) - A227693(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).

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

Vladimir Pletser, Aug 06 2013

nonn

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.

			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

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.

Eric Weisstein's World of Mathematics, Fibonacci Polynomial.

Cf. A006879, A228064, A227693.

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);

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

A229256 Difference between PrimePi(10^n) and its approximation by A229255(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 10, 223, 144, -9998, -58280, 348134, 9517942, 92182430, 404027415, -2717447318, -79612186200, -983858494247, -7964818545554, -31776540093807, 289145607666924, 8243854930562789, 108476952917770938, 885519807642948390, 715407405727600672, -147909423143942345447

Offset: 1

Vladimir Pletser, Sep 17 2013

A229255 provides exact values of pi(10^n) for n=1 to 5 and yields an average relative difference in absolute value of Average(Abs(A229256(n))/pi(10^n)) = 2.05820...*10^-4 for 1<=n<=25.

A229255 provides a better approximation to the distribution of pi(10^n) than: (1) the Riemann function R(10^n) as the sequence of integers nearest to R(10^n), Average(Abs(A057794 (n))/pi(10^n)) =1.219...*10^-2; (2) the functions of the logarithmic integral Li(x) whether as the sequence of integer nearest to (Li(10^n)-Li(3)) (A223166) (Average(Abs(A223167(n))/pi(10^n))= 7.4969...*10^-3), or as Gauss’ approximation to pi(10^n), i.e. the sequence of integer nearest to (Li(10^n)-Li(2)) (A190802) (Average(Abs(A106313(n))/pi(10^n)) =2.0116...*10^-2), or as the sequence of integer nearest to Li(10^n) (A057752) (Average(Abs(A057752 (n))/pi(10^n)) =3.2486...*10^-2).

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..25
C. K. Caldwell, How Many Primes Are There?
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Prime-counting function

Cf. A006880, A229255, A225137, A215663, A057793, A057794, A223166, A223167, A190802, A106313, A057752, A227693, A052435.

a(n) = A006880(n) - A229255(n).

cp's OEIS Frontend

A228115 First differences of A227693.

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A227694 Difference between pi(10^n) and nearest integer to (F[2n+1](S(n)))^2 where pi(10^n) = number of primes <= 10^n (A006880), F[2n+1](x) are Fibonacci polynomials of odd indices [2n+1] and S(n) = Sum_{i=0..2} (C(i)*(log(log(A*(B+n^2))))^(2i)) (see A227693).

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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).

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Formula

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).

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Comments

Examples

References

Links

Crossrefs

Programs

Maple

Formula

A229256 Difference between PrimePi(10^n) and its approximation by A229255(n).

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Formula

A227694 Difference between pi(10^n) and nearest integer to (F[2n+1](S(n)))^2 where pi(10^n) = number of primes <= 10^n (A006880), F[2n+1](x) are Fibonacci polynomials of odd indices [2n+1] and S(n) = Sum_{i=0..2} (C(i)(log(log(A(B+n^2))))^(2i)) (see A227693).

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).

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).