cp's OEIS Frontend

A124318 3-almost primes indexed by semiprimes.

%I A124318 #12 Aug 17 2024 22:29:27
%S A124318 20,28,44,45,66,68,98,99,110,114,147,148,153,165,170,188,207,222,238,
%T A124318 244,245,261,273,284,310,322,343,356,357,363,374,387,388,399,429,438,
%U A124318 465,475,477,494,498,506,531,549,555,590,595,596,602,603,628,639,642
%N A124318 3-almost primes indexed by semiprimes.
%F A124318 a(n) = 3almostprime(semiprime(n)) = A014612(A001358(n)).
%e A124318 a(1) = 3almostprime(semiprime(1)) = 3almostprime(4 = 2^2) = 20 = 2^2 * 5.
%e A124318 a(2) = 3almostprime(semiprime(2)) = 3almostprime(6 = 2 * 3) = 28 = 2^2 * 7.
%e A124318 a(3) = 3almostprime(semiprime(3)) = 3almostprime(9 = 3^2) = 44 = 2^2 * 11.
%e A124318 a(4) = 3almostprime(semiprime(4)) = 3almostprime(10 = 2 * 5) = 45 = 3^2 * 5.
%t A124318 p[k_] := Select[Range[1000], PrimeOmega[#] == k &]; p[3][[Take[p[2], 60]]] (* _Giovanni Resta_, Jun 13 2016 *)
%o A124318 (Python)
%o A124318 from math import isqrt
%o A124318 from sympy import primepi, primerange, integer_nthroot
%o A124318 def A124318(n):
%o A124318     def g(x): return int(x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
%o A124318     def f(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
%o A124318     m, k = n, f(n)+n
%o A124318     while m != k:
%o A124318         m, k = k, f(k)+n
%o A124318     r, k = m, g(m)+m
%o A124318     while r != k:
%o A124318         r, k = k, g(k)+m
%o A124318     return r # _Chai Wah Wu_, Aug 17 2024
%Y A124318 Cf. A124317 Semiprimes indexed by 3-almost primes. A124318 3-almost primes indexed by semiprimes. A124319 semiprime(3almostprime(n)) - 3almostprime(semiprime(n)). A124308 Primes indexed by 5-almost primes. A124309 5-almost primes indexed by primes. A124310 prime(5almostprime(n)) - 5almostprime(prime(n)). 4-almost primes indexed by primes = A124283. 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.
%Y A124318 Cf. A000040, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124317, A124319.
%K A124318 easy,nonn,less
%O A124318 1,1
%A A124318 _Jonathan Vos Post_, Oct 26 2006
%E A124318 a(22)-a(53) from _Giovanni Resta_, Jun 13 2016