cp's OEIS Frontend

A338910 Numbers of the form prime(x) * prime(y) where x and y are both odd.

%I A338910 #28 Apr 22 2025 05:09:06
%S A338910 4,10,22,25,34,46,55,62,82,85,94,115,118,121,134,146,155,166,187,194,
%T A338910 205,206,218,235,253,254,274,289,295,298,314,334,335,341,358,365,382,
%U A338910 391,394,415,422,451,454,466,482,485,514,515,517,527,529,538,545,554
%N A338910 Numbers of the form prime(x) * prime(y) where x and y are both odd.
%F A338910 Numbers m such that A001222(m) = A195017(m) = 2. - _Peter Munn_, Jan 17 2021
%e A338910 The sequence of terms together with their prime indices begins:
%e A338910       4: {1,1}     146: {1,21}    314: {1,37}
%e A338910      10: {1,3}     155: {3,11}    334: {1,39}
%e A338910      22: {1,5}     166: {1,23}    335: {3,19}
%e A338910      25: {3,3}     187: {5,7}     341: {5,11}
%e A338910      34: {1,7}     194: {1,25}    358: {1,41}
%e A338910      46: {1,9}     205: {3,13}    365: {3,21}
%e A338910      55: {3,5}     206: {1,27}    382: {1,43}
%e A338910      62: {1,11}    218: {1,29}    391: {7,9}
%e A338910      82: {1,13}    235: {3,15}    394: {1,45}
%e A338910      85: {3,7}     253: {5,9}     415: {3,23}
%e A338910      94: {1,15}    254: {1,31}    422: {1,47}
%e A338910     115: {3,9}     274: {1,33}    451: {5,13}
%e A338910     118: {1,17}    289: {7,7}     454: {1,49}
%e A338910     121: {5,5}     295: {3,17}    466: {1,51}
%e A338910     134: {1,19}    298: {1,35}    482: {1,53}
%p A338910 q:= n-> (l-> add(i[2], i=l)=2 and andmap(i->
%p A338910     numtheory[pi](i[1])::odd, l))(ifactors(n)[2]):
%p A338910 select(q, [$1..1000])[];  # _Alois P. Heinz_, Nov 23 2020
%t A338910 Select[Range[100],PrimeOmega[#]==2&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]
%o A338910 (Python)
%o A338910 from math import isqrt
%o A338910 from sympy import primepi, primerange
%o A338910 def A338910(n):
%o A338910     def bisection(f,kmin=0,kmax=1):
%o A338910         while f(kmax) > kmax: kmax <<= 1
%o A338910         kmin = kmax >> 1
%o A338910         while kmax-kmin > 1:
%o A338910             kmid = kmax+kmin>>1
%o A338910             if f(kmid) <= kmid:
%o A338910                 kmax = kmid
%o A338910             else:
%o A338910                 kmin = kmid
%o A338910         return kmax
%o A338910     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 A338910     return bisection(f,n,n) # _Chai Wah Wu_, Apr 03 2025
%Y A338910 A338911 is the even instead of odd version.
%Y A338910 A339003 is the squarefree case.
%Y A338910 A001221 counts distinct prime indices.
%Y A338910 A001222 counts prime indices.
%Y A338910 A001358 lists semiprimes, with odd/even terms A046315/A100484.
%Y A338910 A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
%Y A338910 A289182/A115392 list the positions of odd/even terms of A001358.
%Y A338910 A300912 lists semiprimes with relatively prime indices.
%Y A338910 A318990 lists semiprimes with divisible indices.
%Y A338910 A338904 groups semiprimes by weight.
%Y A338910 A338906/A338907 are semiprimes of even/odd weight.
%Y A338910 A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
%Y A338910 A338899, A270650, and A270652 give prime indices of squarefree semiprimes.
%Y A338910 A338909 lists semiprimes with non-relatively prime indices.
%Y A338910 Cf. A005117, A037143, A055684, A056239, A065516, A112798, A195017, A320655, A320732, A320892, A339004.
%K A338910 nonn
%O A338910 1,1
%A A338910 _Gus Wiseman_, Nov 20 2020