cp's OEIS Frontend

A339004 Numbers of the form prime(x) * prime(y) where x and y are distinct and both even.

%I A339004 #22 Jul 27 2025 08:55:09
%S A339004 21,39,57,87,91,111,129,133,159,183,203,213,237,247,259,267,301,303,
%T A339004 321,339,371,377,393,417,427,453,481,489,497,519,543,551,553,559,579,
%U A339004 597,623,669,687,689,703,707,717,749,753,789,791,793,813,817,843,879,917
%N A339004 Numbers of the form prime(x) * prime(y) where x and y are distinct and both even.
%C A339004 The squarefree semiprimes in A332821. - _Peter Munn_, Dec 25 2020
%F A339004 Numbers m such that A001221(m) = A001222(m) = 2 and A195017(m) = -2. - _Peter Munn_, Dec 31 2020
%e A339004 The sequence of terms together with their prime indices begins:
%e A339004      21: {2,4}     267: {2,24}    543: {2,42}
%e A339004      39: {2,6}     301: {4,14}    551: {8,10}
%e A339004      57: {2,8}     303: {2,26}    553: {4,22}
%e A339004      87: {2,10}    321: {2,28}    559: {6,14}
%e A339004      91: {4,6}     339: {2,30}    579: {2,44}
%e A339004     111: {2,12}    371: {4,16}    597: {2,46}
%e A339004     129: {2,14}    377: {6,10}    623: {4,24}
%e A339004     133: {4,8}     393: {2,32}    669: {2,48}
%e A339004     159: {2,16}    417: {2,34}    687: {2,50}
%e A339004     183: {2,18}    427: {4,18}    689: {6,16}
%e A339004     203: {4,10}    453: {2,36}    703: {8,12}
%e A339004     213: {2,20}    481: {6,12}    707: {4,26}
%e A339004     237: {2,22}    489: {2,38}    717: {2,52}
%e A339004     247: {6,8}     497: {4,20}    749: {4,28}
%e A339004     259: {4,12}    519: {2,40}    753: {2,54}
%t A339004 Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&OddQ[Times@@(1+ PrimePi/@First/@FactorInteger[#])]&]
%o A339004 (Python)
%o A339004 from math import isqrt
%o A339004 from sympy import primepi, primerange
%o A339004 def A339004(n):
%o A339004     def bisection(f,kmin=0,kmax=1):
%o A339004         while f(kmax) > kmax: kmax <<= 1
%o A339004         kmin = kmax >> 1
%o A339004         while kmax-kmin > 1:
%o A339004             kmid = kmax+kmin>>1
%o A339004             if f(kmid) <= kmid:
%o A339004                 kmax = kmid
%o A339004             else:
%o A339004                 kmin = kmid
%o A339004         return kmax
%o A339004     def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),1) if a&1^1)
%o A339004     return bisection(f,n,n) # _Chai Wah Wu_, Apr 03 2025
%Y A339004 A338911 is the not necessarily squarefree version.
%Y A339004 A339003 is the odd instead of even version, with not necessarily squarefree version A338910.
%Y A339004 A001358 lists semiprimes, with odd/even terms A046315/A100484.
%Y A339004 A005117 lists squarefree numbers.
%Y A339004 A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
%Y A339004 A289182/A115392 list the positions of odd/even terms in A001358.
%Y A339004 A300912 lists products of pairs of primes with relatively prime indices.
%Y A339004 A318990 lists products of pairs of primes with divisible indices.
%Y A339004 A320656 counts factorizations into squarefree semiprimes.
%Y A339004 A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
%Y A339004 A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
%Y A339004 A338904 groups semiprimes by weight.
%Y A339004 A338906/A338907 list semiprimes of even/odd weight.
%Y A339004 Cf. A000040, A001221, A001222, A056239, A112798, A166237, A195017, A320911, A338901, A338903, A339002.
%Y A339004 Subsequence of A332821.
%K A339004 nonn
%O A339004 1,1
%A A339004 _Gus Wiseman_, Nov 22 2020