A243906 (Number of semiprimes <= n) - (number of primes <= n).

0, -1, -2, -1, -2, -1, -2, -2, -1, 0, -1, -1, -2, -1, 0, 0, -1, -1, -2, -2, -1, 0, -1, -1, 0, 1, 1, 1, 0, 0, -1, -1, 0, 1, 2, 2, 1, 2, 3, 3, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 5, 4, 4, 3, 4, 4, 4, 5, 5, 4, 4, 5, 5, 4, 4, 3, 4, 4, 4, 5, 5, 4, 4, 4, 5, 4, 4, 5, 6, 7, 7, 6, 6, 7, 7, 8

Offset

1,3

Comments

We know from the asymptotic formulas (see Landau) that the sequence is almost always positive.

References

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig, 1909; third edition : Chelsea, New York (1974).

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..5195

Formula

a(n) = A072000(n) - A000720(n). - Michel Marcus, Dec 20 2022

Maple

g:= proc(n) if isprime(n) then -1 elif numtheory:-bigomega(n) = 2 then 1 else 0 fi end proc:
ListTools:-PartialSums(map(g, [$1..100])); # Robert Israel, Dec 20 2022

Mathematica

Accumulate[Table[Which[PrimeQ[n], -1, PrimeOmega[n]==2, 1, True, 0], {n, 1000}]] (* Harvey P. Dale, Jun 15 2014 *)

Prog

(PARI) a(n) = #select(x->(bigomega(x) == 2), [1..n]) - primepi(n); \\ Michel Marcus, Dec 20 2022
(Python)
from math import isqrt
from sympy import prime, primepi
def A243906(n): return int(sum(primepi(n//prime(k))-k+1 for k in range(1, primepi(isqrt(n))+1)))-primepi(n) # Chai Wah Wu, Jul 23 2024

Crossrefs

Cf. A243903, A276176.

Cf. A000720, A072000.

Sequence in context: A232506 A133099 A006571 * A100889 A206828 A327164

Adjacent sequences: A243903 A243904 A243905 * A243907 A243908 A243909

Keyword

sign

Author

N. J. A. Sloane, Jun 14 2014

Status

approved