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A124283 4-almost primes indexed by primes.

%I A124283 #9 Sep 10 2024 00:24:44
%S A124283 24,36,54,60,90,104,136,150,189,225,232,294,308,328,344,375,441,459,
%T A124283 488,510,516,550,570,621,676,708,714,738,748,776,852,860,884,910,999,
%U A124283 1014,1060,1096,1112,1161,1197,1206,1256,1274,1284,1290,1356,1432,1450,1482
%N A124283 4-almost primes indexed by primes.
%C A124283 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.
%H A124283 Giovanni Resta, <a href="/A124283/b124283.txt">Table of n, a(n) for n = 1..10000</a>
%F A124283 a(n) = 4almostprime(prime(n)) = A014613(A000040(n)).
%e A124283 a(1) = 4almostprime(prime(1)) = 4almostprime(2) = 24.
%e A124283 a(2) = 4almostprime(prime(2)) = 4almostprime(3) = 36.
%e A124283 a(3) = 4almostprime(prime(3)) = 4almostprime(5) = 54.
%o A124283 (Python)
%o A124283 from math import isqrt
%o A124283 from sympy import prime, primepi, integer_nthroot, primerange
%o A124283 def A124283(n):
%o A124283     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)))
%o A124283     def bisection(f,kmin=0,kmax=1):
%o A124283         while f(kmax) > kmax: kmax <<= 1
%o A124283         while kmax-kmin > 1:
%o A124283             kmid = kmax+kmin>>1
%o A124283             if f(kmid) <= kmid:
%o A124283                 kmax = kmid
%o A124283             else:
%o A124283                 kmin = kmid
%o A124283         return kmax
%o A124283     return bisection(f,n,n) # _Chai Wah Wu_, Sep 09 2024
%Y A124283 Cf. A000040, A014613, A114414, A122824, A124269, A124270, A124282, A124284.
%K A124283 easy,nonn
%O A124283 1,1
%A A124283 _Jonathan Vos Post_, Oct 24 2006
%E A124283 a(17)-a(50) from _Giovanni Resta_, Jun 13 2016