cp's OEIS Frontend

A339003 Numbers of the form prime(x) * prime(y) where x and y are distinct and both odd.

%I A339003 #18 Apr 22 2025 05:08:35
%S A339003 10,22,34,46,55,62,82,85,94,115,118,134,146,155,166,187,194,205,206,
%T A339003 218,235,253,254,274,295,298,314,334,335,341,358,365,382,391,394,415,
%U A339003 422,451,454,466,482,485,514,515,517,527,538,545,554,566,614,626,635,649
%N A339003 Numbers of the form prime(x) * prime(y) where x and y are distinct and both odd.
%C A339003 The squarefree semiprimes in A332822. - _Peter Munn_, Dec 25 2020
%F A339003 Numbers m such that A001221(m) = A001222(m) = A195017(m) = 2. - _Peter Munn_, Dec 31 2020
%e A339003 The sequence of terms together with their prime indices begins:
%e A339003      10: {1,3}     187: {5,7}     358: {1,41}    527: {7,11}
%e A339003      22: {1,5}     194: {1,25}    365: {3,21}    538: {1,57}
%e A339003      34: {1,7}     205: {3,13}    382: {1,43}    545: {3,29}
%e A339003      46: {1,9}     206: {1,27}    391: {7,9}     554: {1,59}
%e A339003      55: {3,5}     218: {1,29}    394: {1,45}    566: {1,61}
%e A339003      62: {1,11}    235: {3,15}    415: {3,23}    614: {1,63}
%e A339003      82: {1,13}    253: {5,9}     422: {1,47}    626: {1,65}
%e A339003      85: {3,7}     254: {1,31}    451: {5,13}    635: {3,31}
%e A339003      94: {1,15}    274: {1,33}    454: {1,49}    649: {5,17}
%e A339003     115: {3,9}     295: {3,17}    466: {1,51}    662: {1,67}
%e A339003     118: {1,17}    298: {1,35}    482: {1,53}    685: {3,33}
%e A339003     134: {1,19}    314: {1,37}    485: {3,25}    694: {1,69}
%e A339003     146: {1,21}    334: {1,39}    514: {1,55}    697: {7,13}
%e A339003     155: {3,11}    335: {3,19}    515: {3,27}    706: {1,71}
%e A339003     166: {1,23}    341: {5,11}    517: {5,15}    713: {9,11}
%t A339003 Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]
%o A339003 (Python)
%o A339003 from math import isqrt
%o A339003 from sympy import primepi, primerange
%o A339003 def A339003(n):
%o A339003     def bisection(f,kmin=0,kmax=1):
%o A339003         while f(kmax) > kmax: kmax <<= 1
%o A339003         kmin = kmax >> 1
%o A339003         while kmax-kmin > 1:
%o A339003             kmid = kmax+kmin>>1
%o A339003             if f(kmid) <= kmid:
%o A339003                 kmax = kmid
%o A339003             else:
%o A339003                 kmin = kmid
%o A339003         return kmax
%o A339003     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)
%o A339003     return bisection(f,n,n) # _Chai Wah Wu_, Apr 03 2025
%Y A339003 A338910 is the not necessarily squarefree version.
%Y A339003 A339004 is the even instead of odd version.
%Y A339003 A001358 lists semiprimes, with odd and even terms A046315 and A100484.
%Y A339003 A005117 lists squarefree numbers.
%Y A339003 A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
%Y A339003 A289182/A115392 list the positions of odd/even terms of A001358.
%Y A339003 A300912 lists products of two primes of relatively prime index.
%Y A339003 A320656 counts factorizations into squarefree semiprimes.
%Y A339003 A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
%Y A339003 A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
%Y A339003 A338904 groups semiprimes by weight.
%Y A339003 A338906/A338907 list semiprimes of even/odd weight.
%Y A339003 A339002 lists products of two distinct primes of non-relatively prime index.
%Y A339003 A339005 lists products of two distinct primes of divisible index.
%Y A339003 Cf. A001221, A001222, A056239, A112798, A166237, A195017, A318990, A320911, A338901, A338903, A338911.
%Y A339003 Subsequence of A332822.
%K A339003 nonn
%O A339003 1,1
%A A339003 _Gus Wiseman_, Nov 21 2020