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A124317 Semiprimes indexed by 3-almost primes.

%I A124317 #14 Aug 17 2024 22:29:18
%S A124317 22,34,51,57,82,85,87,123,133,134,146,158,201,205,209,214,221,226,237,
%T A124317 295,305,309,321,327,341,361,365,371,394,395,413,447,478,481,497,501,
%U A124317 529,533,543,545,551,554,559,583,597,614,623,635,689,699,734,763,766
%N A124317 Semiprimes indexed by 3-almost primes.
%C A124317 Note that a(10)-a(9) = a(30)-a(29) = 1, achieving the minimum possible, due to a combination of the appropriate semiprime gap (A065516) and 3-almost prime gap (A114403).
%H A124317 Giovanni Resta, <a href="/A124317/b124317.txt">Table of n, a(n) for n = 1..5000</a>
%F A124317 a(n) = semiprime(3almostprime(n)) = A001358(A014612(n)).
%e A124317 a(1) = semiprime(3almostprime(1)) = semiprime(8 = 2^3) = 22 = 2 * 11.
%e A124317 a(2) = semiprime(3almostprime(2)) = semiprime(12 = 2^2 * 3) = 34 = 2 * 17.
%e A124317 a(3) = semiprime(3almostprime(3)) = semiprime(18 = 2 * 3^2) = 51 = 3 * 17.
%t A124317 p[k_] := Select[Range[1000], PrimeOmega[#] == k &]; p[2][[Take[p[3], 60]]] (* _Giovanni Resta_, Jun 13 2016 *)
%o A124317 (Python)
%o A124317 from math import isqrt
%o A124317 from sympy import primepi, primerange, integer_nthroot
%o A124317 def A124317(n):
%o A124317     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 A124317     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 A124317     m, k = n, f(n)+n
%o A124317     while m != k:
%o A124317         m, k = k, f(k)+n
%o A124317     r, k = m, g(m)+m
%o A124317     while r != k:
%o A124317         r, k = k, g(k)+m
%o A124317     return r # _Chai Wah Wu_, Aug 17 2024
%Y A124317 Cf. 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 A124317 Cf. A000040, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124318, A124319.
%K A124317 easy,nonn,less
%O A124317 1,1
%A A124317 _Jonathan Vos Post_, Oct 26 2006
%E A124317 Data corrected by _Giovanni Resta_, Jun 13 2016