A124283 4-almost primes indexed by primes.

24, 36, 54, 60, 90, 104, 136, 150, 189, 225, 232, 294, 308, 328, 344, 375, 441, 459, 488, 510, 516, 550, 570, 621, 676, 708, 714, 738, 748, 776, 852, 860, 884, 910, 999, 1014, 1060, 1096, 1112, 1161, 1197, 1206, 1256, 1274, 1284, 1290, 1356, 1432, 1450, 1482

Offset: 1

Author

Jonathan Vos Post, Oct 24 2006

Keywords

• easy
• nonn

Comments

Primes indexed by 4-almost primes = A124282. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

Examples

			a(1) = 4almostprime(prime(1)) = 4almostprime(2) = 24.
a(2) = 4almostprime(prime(2)) = 4almostprime(3) = 36.
a(3) = 4almostprime(prime(3)) = 4almostprime(5) = 54.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A014613, A114414, A122824, A124269, A124270, A124282, A124284.

Programs

Python

from math import isqrt
from sympy import prime, primepi, integer_nthroot, primerange
def A124283(n):
    def f(x): return int(prime(n)+x-sum(primepi(x//(k*m*r))-c for a, k in enumerate(primerange(integer_nthroot(x, 4)[0]+1)) for b, m in enumerate(primerange(k, integer_nthroot(x//k, 3)[0]+1), a) for c, r in enumerate(primerange(m, isqrt(x//(k*m))+1), b)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

Formula

a(n) = 4almostprime(prime(n)) = A014613(A000040(n)).

Extensions

a(17)-a(50) from Giovanni Resta, Jun 13 2016