A058290 -id:A058290 - cp's OEIS frontend

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

-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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions