A099319 Numerators of an approximation of Riemann to pi(n).

0, 1, 3, 9, 3, 7, 4, 14, 61, 16, 35, 19, 41, 22, 22, 179, 97, 103, 109, 115, 115, 115, 121, 127, 65, 133, 45, 137, 143, 149, 155, 811, 817, 817, 817, 817, 847, 877, 877, 877, 907, 937, 967, 997, 997, 997, 1027, 1057, 268, 1087, 1087, 1087, 1117, 1147, 1147, 1147, 1147

Offset

1,3

Comments

Edwards, p. 22, calls this J(n).

References

J. C. Lagarias and A. M. Odlyzko, Computing pi(x): an analytic method, J. Algorithms, 8 (1987), 173-191.

H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974.

Links

T. D. Noe, Table of n, a(n) for n=1..10000

Formula

See Maple code.

Example

0, 1/2, 3/2, 9/4, 3, 7/2, 4, 14/3, 61/12, 16/3, 35/6, 19/3,... = A099319/A099320.

Maple

f:=proc(n) local i, m, p, t1, t2; t1:=0; for i from 1 to n do p:=ithprime(i); if p > n then break; fi; for m from 1 to n do if p^m > n then break; fi; if n = p^m then t2:=1/(2*m) else t2:=1/m; fi; t1:=t1+t2; od; od; t1; end;

Mathematica

f[n_] := Module[{i, m, p, t1, t2}, t1 = 0; For[i = 1, i <= n, i++, p = Prime[i]; If[p > n, Break[]]; For[m = 1, m <= n, m++,  If[p^m > n, Break[]]; If[n == p^m, t2 = 1/(2m), t2 = 1/m]; t1 = t1 + t2]]; t1];
Table[f[n] // Numerator, {n, 1, 100}] (* Jean-François Alcover, Apr 02 2023, after Maple code _)

Crossrefs

Cf. A099320

Sequence in context: A224233 A021258 A200607 * A231828 A268580 A179802

Adjacent sequences: A099316 A099317 A099318 * A099320 A099321 A099322

Keyword

nonn,frac

Author

N. J. A. Sloane, Nov 17 2004

Status

approved