A057794 (Integer nearest R(10^n)) - pi(10^n), where pi(x) is the number of primes <= x, R(x) = Sum_{ k>=1 } (mu(k)/k * li(x^(1/k))) and li(x) is the Cauchy principal value of the integral from 0 to x of dt/log(t).
1, 1, 0, -2, -5, 29, 88, 97, -79, -1828, -2318, -1476, -5773, -19200, 73218, 327052, -598255, -3501366, 23884333, -4891825, -86432204, -127132665, 1033299853, -1658989719, -1834784714, -17149335456, -17535487934, -174760519827
Offset: 1
Keywords
References
- John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, page 146.
- M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 90.
Links
- Anatolii A. Karatsuba and Ekatherina A. Karatsuba, The "problem of remainders" in theoretical physics: "physical zeta" function, 6th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, 14-23 September 2010, Belgrade, Serbia. [From Internet Archive Wayback Machine]
- Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
- Eric Weisstein's World of Mathematics, Prime Counting Function.
- Eric Weisstein's World of Mathematics, Riemann Prime Number Formula.
- Eric Weisstein's World of Mathematics, Gram Series.
Programs
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Mathematica
R[x_] := Sum[N[LogIntegral[x^(1/k)]*MoebiusMu[k]/k, 36], {k, 1, 1000}]; a[n_] := Abs[Round[R[10^n]-PrimePi[10^n]]] gram[x_] := 1+Sum[N[Log[x]^k/(k*k!*Zeta[k+1]), 100], {k, 1, 1000}]; a[n_] := Abs[Round[gram[10^n]-PrimePi[10^n]]] (* From version 7 on : *) a[n_] := Round[RiemannR[10^n]-PrimePi[10^n]] (* Jean-François Alcover, Sep 17 2012 *) -
PARI
A057794=vector(#A006880,i,round(1+suminf(k=1, log(10^i)^k/(k*k!*zeta(k+1)))-A006880[i])) \\ - M. F. Hasler, Feb 26 2008
Extensions
First term corrected by David Baugh, Nov 15 2002
Signs added by M. F. Hasler, Feb 26 2008
The value of a(23) is not known at present, I believe. - N. J. A. Sloane, Mar 17 2008
Last two terms a(23) and a(24), with Pi(10^n) for n=23 and 24 from A006880, from Vladimir Pletser, Feb 27 2013
Terms a(25)-a(28) obtained using A006880. - Eduard Roure Perdices, Apr 13 2021
Comments