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A124319 Semiprime(3almostprime(n))-3almostprime(semiprime(n)). Commutator[A001358, A014612] at n.

%I A124319 #11 Aug 17 2024 22:29:35
%S A124319 2,6,7,12,16,17,-11,24,23,20,-1,10,48,40,39,26,14,4,-1,51,60,48,48,43,
%T A124319 31,39,22,15,37,32,39,60,90,82,68,63,64,58,66,51,53,48,28,34,42,24,28,
%U A124319 39,87,96,106,124,124,135,131,131,88,91,72,96,103,83,83,81,91
%N A124319 Semiprime(3almostprime(n))-3almostprime(semiprime(n)). Commutator[A001358, A014612] at n.
%e A124319 a(1) = semiprime(3almostprime(1)) - 3almostprime(semiprime(1)) = 22 - 20 = 2.
%e A124319 a(2) = semiprime(3almostprime(2)) - 3almostprime(semiprime(2)) = 34 - 28 = 6.
%e A124319 a(3) = semiprime(3almostprime(3)) - 3almostprime(semiprime(3)) = 51 - 44 = 7.
%e A124319 a(4) = semiprime(3almostprime(4)) - 3almostprime(semiprime(4)) = 57 - 45 = 12.
%e A124319 a(7) = semiprime(3almostprime(7)) - 3almostprime(semiprime(7)) = 87 - 98 = -11, which is the first negative value in the commutators we have seen in these related set of sequences, exposing an incorrect assumption.
%t A124319 p[k_] := p[k] = Select[Range[1000], PrimeOmega[#] == k &]; p[2][[ Take[p[3], 70]]] - p[3][[Take[p[2], 70]]] (* _Giovanni Resta_, Jun 13 2016 *)
%o A124319 (Python)
%o A124319 from math import isqrt
%o A124319 from sympy import primepi, primerange, integer_nthroot
%o A124319 def A124319(n):
%o A124319     def f(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 A124319     def g(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
%o A124319     def A001358(n):
%o A124319         m, k = n, g(n)+n
%o A124319         while m != k:
%o A124319             m, k = k, g(k)+n
%o A124319         return m
%o A124319     m, k = n, f(n)+n
%o A124319     while m != k:
%o A124319         m, k = k, f(k)+n
%o A124319     r, k = (p:=A001358(n)), f(p)+p
%o A124319     while r != k:
%o A124319         r, k = k, f(k)+p
%o A124319     return A001358(m)-r # _Chai Wah Wu_, Aug 17 2024
%Y A124319 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 A124319 Cf. A000040, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124317, A124318.
%K A124319 easy,sign,less
%O A124319 1,1
%A A124319 _Jonathan Vos Post_, Oct 26 2006
%E A124319 a(18) corrected and a(22)-a(65) from _Giovanni Resta_, Jun 13 2016