A228066 -id:A228066 - 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).

A228068 Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to Li(10^n) - Li(2) (see A228067).

Original entry on oeis.org

-1, -3, -5, -7, -21, -92, -209, -415, -947, -1403, -8484, -26675, -70708, -205919, -737729, -2162013, -4741957, -13992966, -77928220, -122866869, -374649610, -1334960954, -5317831008, -9896721062, -38014073661

Offset: 1

Vladimir Pletser, Aug 06 2013

sign

The sequence A006879(n) is always < A228067(n) for 1 <= n <= 25.
The sequence (A228067) yields an average relative difference in absolute value, i.e., average(abs(A228068(n))/A006879(n) = 0.0175492... for 1 <= n <= 25.
Note that A190802(n) = (Li(10^n) - Li(2)) is not defined for n=0. Its value is set arbitrarily to 0.

Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Logarithmic Integral

Cf. A006880, A006879, A228067, A228066.

a(n) = A006879(n) - A228067(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

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).

A228064 Difference between the number of primes with n digits (A006879) and the nearest integer 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 A228063).

Original entry on oeis.org

0, 0, 0, -2, -8, 121, 2645, 27243, 209322, 1179803, 2299680, -61020043, -1269344630, -17189254160, -195686557968, -1996027658061, -18568445615842, -156279759410226, -1137747666182762, -6044328439309231, 1630706099481822, 705861452287757875

Offset: 1

Vladimir Pletser, Aug 06 2013

sign

A228063 provides exactly the 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(A228064(n))/A006879(n) = 0.00473860... 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) 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.

a(n) = A006879(n) - A228063(n).

A228065 Difference of consecutive integers nearest to (10^n)/log(10^n) (A057834).

Original entry on oeis.org

4, 18, 123, 941, 7600, 63696, 548039, 4808260, 42826261, 386039540, 3513837172, 32243075171, 297881471562, 2768030763779, 25850862018051, 242481085729315, 2283239371770773, 21572797793887019, 204448571890127322, 1942896366409284492

Offset: 1

Vladimir Pletser, Aug 06 2013

nonn

This sequence gives an approximation of the number of primes with n digits (A006879); see A228066.

Note that A057834(n) = (10^n)/log(10^n) is not defined for n=0. Its value is set arbitrarily to 0.

			For n = 1, A057834(1) - A057834(0) = 4-0 = 4.

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

Cf. A006879, A057834, A228066.

a(n) = A057834(n) - A057834(n-1).

cp's OEIS Frontend

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

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A228068 Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to Li(10^n) - Li(2) (see A228067).

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

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A228064 Difference between the number of primes with n digits (A006879) and the nearest integer 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 A228063).

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A228065 Difference of consecutive integers nearest to (10^n)/log(10^n) (A057834).

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Links

Crossrefs

Formula

A228064 Difference between the number of primes with n digits (A006879) and the nearest integer 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 A228063).