A071903 Number of x less than or equal to n and divisible only by primes congruent to 3 mod 4 (i.e., in A004614).

1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 22

Offset: 0

Author

Benoit Cloitre, Jun 12 2002

Keywords

• easy
• nonn

References

• Landau, "Handbuch der Lehre von der Verteilung der Primzahlen", vol. 2, Teubner, Leipzig; third edition: Chelsea, New York (1974), pp. 641-669.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
• E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

Crossrefs

Cf. A002145, A004614.

Programs

Mathematica

With[{s = {1}~Join~Select[Range@ 80, AllTrue[FactorInteger[#][[All, 1]], Mod[#, 4] == 3 &] &]}, Table[LengthWhile[s, # <= n &], {n, Max@ s}]] (* Michael De Vlieger, Jul 30 2017 *)

PARI

for(n=1,100,print1(sum(i=1,n,if(sumdiv(i,d,isprime(d)*(d-3)%4),0,1)),","))

Formula

a(n) = Card{ k | A004614(k) <= n }.

Asymptotically: a(n) ~ sqrt(2)*A*n/(Pi*sqrt(log(n))) where A = Product_{k>0} ((1-A002145(k)^(-2))^(-1/2)).