A057793 -id:A057793 - cp's OEIS frontend

A228113 First differences of A057793.

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).

A058289 Integer nearest 10^n/(log(10^n) - 1.08366).

Original entry on oeis.org

-1, 8, 28, 172, 1231, 9588, 78543, 665140, 5768004, 50917519, 455743004, 4124599869, 37668527415, 346621096885, 3210012022164, 29890794226982, 279660033612131, 2627410589445923, 24775244142175635, 234381646366460804

Offset: 0

Robert G. Wilson v, Dec 07 2000

sign

"Adrien-Marie Legendre in 1778 published his work 'Essai sur la théorie des nombres' where he proposed a modified form of the first approximation, pi(n) ~ n/ln n." (Gullberg)

Jan Gullberg, "Mathematics, From the Birth of Numbers," W. W. Norton and Company, NY and London, 1997, page 80.

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A006880, A057754, A057793, A057834, A058290.

Table[ Round[ 10^n /(Log[10^n] - 1.08366) ], {n, 0, 22} ]

{ default(realprecision, 1000); t=log(10); for (n=0, 500, write("b058289.txt", n, " ", round(10^n/(n*t - 1.08366))); ); } \\ Harry J. Smith, Jun 22 2009

Corrected some terms. - Harry J. Smith, Jun 22 2009

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).

A226945 Integer nearest f(10^n), where f(x) = Sum of ( mu(k) * H(k)/k^(3/2) * Integral Log(x^(1/k)) ) for k = 1 to infinity, where H(k) is the harmonic number sum_{i=1..k} 1/i.

Original entry on oeis.org

4, 25, 168, 1226, 9585, 78521, 664652, 5761512, 50847348, 455050385, 4118051652, 37607908133, 346065524108, 3204941711340, 29844570436484, 279238341185832, 2623557156537070, 24739954282695698, 234057667295619287, 2220819602542218793

Offset: 1

Arkadiusz Wesolowski, Aug 31 2013

nonn

The sequence gives exactly the values of pi(10^n) for n = 1 to 3.

A228724 gives the difference between A006880 and this sequence.

David Baugh, Table of n, a(n) for n = 1..100
Chris K. Caldwell, How Many Primes Are There?
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Prime Number Theorem

Cf. A006880, A057834, A226744, A057754, A190802, A057793, A201542, A228724.

f[n_Integer] := Sum[N[MoebiusMu[k]*HarmonicNumber[k]/k^(3/2)*LogIntegral[n^(1/k)], 50], {k, 5!}]; Table[Round[f[10^n]], {n, 20}]

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).

A163516 a(n) = floor( Sum_{x=2..n} x/log(x) ).

Original entry on oeis.org

0, 2, 5, 8, 11, 14, 18, 22, 26, 30, 35, 40, 45, 50, 56, 61, 67, 74, 80, 87, 94, 101, 108, 116, 123, 131, 140, 148, 157, 165, 175, 184, 193, 203, 213, 223, 233, 243, 254, 265, 276, 287, 299, 310, 322, 334, 346, 359, 371, 384, 397, 410, 424, 437, 451, 465, 479, 493

Offset: 1

Cino Hilliard, Jul 30 2009

nonn

a(n) closely approximates the number of primes < n^2, that is, A038107(n) = Pi(n^2).
In fact, the sum is as good as Li(n^2). For n = 10^9,
  a(n)    = 24739954333817884.
  Pi(n^2) = 24739954287740860 = A006880(18).
  Li(n^2) = 24739954309690415 = A057754(18) = A089896(18).
  R(n^2)  = 24739954284239494 = A057793(18).
Now x/(log(x)-1) is a much better approximation of Pi(x) than x/log(x).
  10^18/(log(10^18)-1) = 24723998785919976 and
  10^18/log(10^18)     = 24127471216847323.
Ironically though, a(n) = Sum_{x=2..n} x/(log(x)-1) is far from Pi(n^2), see A058290.

			For n = 10, floor(Sum_{x=2..n} x/log(x)) = 30, the 10th term.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Table[Floor[Sum[j/Log[j], {j, 2, n}]], {n,1,50}] (* G. C. Greubel, Jul 27 2017 *)
Join[{0},Floor[Accumulate[Table[x/Log[x],{x,2,60}]]]] (* Harvey P. Dale, May 22 2021 *)

nthsum(n) = for(j=1,n,print1(floor(sum(x=2,j,x/log(x)))","));

a(10^n) = A163521(n).

Offset corrected, definition detailed, 7 references to other sequences added by R. J. Mathar, Aug 29 2009

A353055 Successive records of function f(x) = log(abs(pi(x) - R(x)))/log(x) where pi(x) is the number of primes <= x and R(x) is Riemann's prime counting function.

Original entry on oeis.org

2, 4, 7, 10, 19, 47, 58, 73, 109, 113, 1109, 1123, 1129, 1307, 1321, 1327, 1418, 1419, 1420, 1421, 1422, 5379, 5380, 7449, 7450, 10343, 11676, 11761, 11762, 11763, 11764, 11765, 11766, 11767, 11768, 11769, 11770, 11771, 11772, 11773, 11774, 11775, 11776, 19360, 19361, 19362, 19363, 19364, 19365, 19366, 19367, 19368, 19369, 19370, 19371, 19372

Offset: 1

Artur Jasinski, Apr 20 2022

nonn

Two possibilities:
(1) this sequence is finite;
(2) this sequence is infinite.
In case (1) there exists a maximal integer x_max such that J = f(x_max) = log(abs(pi(x_max) - R(x_max)))/log(x_max).
In case (2) there exists a real constant J such that lim_{x->oo} f(x) = J.
Then for every positive integer x, abs((R(x) - pi(x))/x^J) <= 1.
According to actual computations biggest x = 1090696 with log(-85020 + R(1090696))/log(1090696) = 0.27835121240340474... and no more new terms up to x 3000000. Follow this:
0.27835121240340474... <= J.
J < 1/2 = limit((log(x) - 2*log((8*Pi)/log(x)))/(2*log(x)), x -> infinity) proof follow Lowell Schoenfeld 1976 proof on upper limit of Chebyshev function psi(x).
Constant J can be used to measure best proved upper limits of asymptotical behavior of function pi(x) when x->infinity. If J is smaller upper bound is better.

			x      f(x)                comment
1     -infinity            a(1)
2     -0.8862754573970588  a(2)
3     -4.883591467412115   removed because f(3) < f(2)
4     -0.614424415865155   a(3)
5     -1.0695141714266385  removed because f(5) < f(4)
...    ...

Lowell Schoenfeld, Sharper Bounds for the Chebyshev Functions theta(x) and psi(x). II, Mathematics of Computation Vol. 30, No. 134 (Apr., 1976), pp. 337-360.

Cf. A057793, A057794, A215663, A216709, A226473.

gg = {1}; imax = -1000; Do[
kk = Log[Abs[PrimePi[x] - RiemannR[x]]]/Log[x];
If[kk > imax, AppendTo[gg, x]; imax = kk], {x, 2, 20000}]; gg

cp's OEIS Frontend

A228113 First differences of A057793.

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A058289 Integer nearest 10^n/(log(10^n) - 1.08366).

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

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A226945 Integer nearest f(10^n), where f(x) = Sum of ( mu(k) * H(k)/k^(3/2) * Integral Log(x^(1/k)) ) for k = 1 to infinity, where H(k) is the harmonic number sum_{i=1..k} 1/i.

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

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A163516 a(n) = floor( Sum_{x=2..n} x/log(x) ).

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A353055 Successive records of function f(x) = log(abs(pi(x) - R(x)))/log(x) where pi(x) is the number of primes <= x and R(x) is Riemann's prime counting function.

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