This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A060208 #42 Jan 05 2025 19:51:36 %S A060208 -1,0,1,0,2,1,2,2,1,0,2,1,3,3,2,1,3,3,4,4,3,2,4,3,3,3,2,2,4,3,4,4,4,3, %T A060208 3,2,3,3,3,2,4,3,5,5,4,4,6,6,5,5,4,3,5,4,3,3,2,2,4,4,6,6,6,5,5,4,6,6, %U A060208 5,4,6,6,8,8,7,6,6,6,7,7,7,6,8,7,7,7,6,6,8,7,6,6,6,6,6,5,6,6,5,4,6,6,8,8,8,7,9,9,11,11,11,10,12,11,10,10,9,9,9,8,7,7 %N A060208 a(n) = 2*pi(n) - pi(2*n), where pi(i) = A000720(i). %C A060208 Rosser & Schoenfeld show 2*pi(x) > pi(2*x) for x > 10. - _N. J. A. Sloane_, Jul 03 2013, corrected Jul 09 2015 %D A060208 J. Barkley Rosser and Lowell Schoenfeld, Abstracts of Scientific Communications, Internat. Congress Math., Moscow, 1966, Section 3, Theory of Numbers. %D A060208 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235. %D A060208 Sanford Segal, On Pi(x+y)<=Pi(x)+Pi(y). Transactions American Mathematical Society, 104 (1962), 523-527. %H A060208 T. D. Noe, <a href="/A060208/b060208.txt">Table of n, a(n) for n = 1..10000</a> %H A060208 Eugene Ehrhart, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/26-3/ehrhart.pdf">On prime numbers</a>, Fibonacci Quarterly 26:3 (1988), pp. 271-274. Shows a(n)>0 for n>10. %H A060208 E. Labos, <a href="/A060208/a060208.html">Illustration</a> %F A060208 a(n) = Mod[2*PrimePi[n], PrimePi[2n]] = 2*A000720(n) - A000720(2n) for n>1. %F A060208 a(n) ~ 2n log 2 / (log n)^2, by the prime number theorem. - _N. J. A. Sloane_, Mar 12 2007 %F A060208 a(n) = -A047886(n,n) (see A212210 to A212213). - _Reinhard Zumkeller_, Apr 15 2008 %e A060208 n=100, pi(100)=25, pi(200)=46, 2pi(100)-pi(2*100) =4=a(100) %t A060208 f[n_] := 2 PrimePi[n] - PrimePi[2 n]; Array[f, 122] (* _Robert G. Wilson v_, Aug 12 2011 *) %o A060208 (PARI) a(n)=2*primepi(n)-primepi(2*n) \\ _Charles R Greathouse IV_, Jul 02 2013 %o A060208 (Magma) [2*#PrimesUpTo(n) -#PrimesUpTo(2*n): n in [1..200]]; // _G. C. Greubel_, Aug 01 2024 %o A060208 (SageMath) [2*prime_pi(n) -prime_pi(2*n) for n in range(1,201)] # _G. C. Greubel_, Aug 01 2024 %Y A060208 Cf. A000720, A007097, A033844, A060207, A212210, A212211, A212212, A212213. %K A060208 sign %O A060208 1,5 %A A060208 _Labos Elemer_, Mar 19 2001 %E A060208 Edited by _N. J. A. Sloane_, Jul 03 2013