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A124283
4-almost primes indexed by primes.
8
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
1,1
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.
LINKS
FORMULA
a(n) = 4almostprime(prime(n)) = A014613(A000040(n)).
EXAMPLE
a(1) = 4almostprime(prime(1)) = 4almostprime(2) = 24.
a(2) = 4almostprime(prime(2)) = 4almostprime(3) = 36.
a(3) = 4almostprime(prime(3)) = 4almostprime(5) = 54.
PROG
(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
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Oct 24 2006
EXTENSIONS
a(17)-a(50) from Giovanni Resta, Jun 13 2016
STATUS
approved