cp's OEIS Frontend

A046388 Odd numbers of the form p*q where p and q are distinct primes.

%I A046388 #94 Sep 10 2024 14:22:23
%S A046388 15,21,33,35,39,51,55,57,65,69,77,85,87,91,93,95,111,115,119,123,129,
%T A046388 133,141,143,145,155,159,161,177,183,185,187,201,203,205,209,213,215,
%U A046388 217,219,221,235,237,247,249,253,259,265,267,287,291,295,299,301,303
%N A046388 Odd numbers of the form p*q where p and q are distinct primes.
%C A046388 These are the odd squarefree semiprimes.
%C A046388 These numbers k have the property that k is a Fermat pseudoprime for at least two bases 1 < b < k - 1. That is, b^(k - 1) == 1 (mod k). See sequence A175101 for the number of bases. - _Karsten Meyer_, Dec 02 2010
%H A046388 Amiram Eldar, <a href="/A046388/b046388.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%F A046388 Sum_{n>=1} 1/a(n)^s = (1/2)*(P(s)^2 - P(2*s)) + 1/4^s - P(s)/2^s, for s>1, where P is the prime zeta function. - _Amiram Eldar_, Nov 21 2020
%t A046388 max = 300; A046388 = Sort@Flatten@Table[Prime[m] Prime[n], {n, 3, Ceiling[PrimePi[max/3]]}, {m, 2, n - 1}]; Select[A046388, # < max &] (* _Alonso del Arte_ based on _Robert G. Wilson v_'s program for A006881, Oct 24 2011 *)
%o A046388 (Haskell)
%o A046388 a046388 n = a046388_list !! (n-1)
%o A046388 a046388_list = filter ((== 2) . a001221) a056911_list
%o A046388 -- _Reinhard Zumkeller_, Jan 02 2014
%o A046388 (PARI) isok(n) = (n % 2) && (bigomega(n) == 2) && (omega(n)==2); \\ _Michel Marcus_, Feb 05 2015
%o A046388 (Python)
%o A046388 from sympy import factorint
%o A046388 def ok(n):
%o A046388     if n < 2 or n%2 == 0: return False
%o A046388     f = factorint(n)
%o A046388     return len(f) == 2 and sum(f.values()) == 2
%o A046388 print([k for k in range(304) if ok(k)]) # _Michael S. Branicky_, May 03 2022
%o A046388 (Python)
%o A046388 from math import isqrt
%o A046388 from sympy import primepi, primerange
%o A046388 def A046388(n):
%o A046388     if n == 1: return 15
%o A046388     def f(x): return int(n-1+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+1)))
%o A046388     def bisection(f,kmin=0,kmax=1):
%o A046388         while f(kmax) > kmax: kmax <<= 1
%o A046388         while kmax-kmin > 1:
%o A046388             kmid = kmax+kmin>>1
%o A046388             if f(kmid) <= kmid:
%o A046388                 kmax = kmid
%o A046388             else:
%o A046388                 kmin = kmid
%o A046388         return kmax
%o A046388     return bisection(f,n,n) # _Chai Wah Wu_, Sep 10 2024
%Y A046388 Intersection of A005117 and A046315, or equally, of A005408 and A006881, or of A001358 and A056911.
%Y A046388 Union of A080774 and A190299, which the latter is the union of A131574 and A016105.
%Y A046388 Subsequence of A024556 and of A225375.
%Y A046388 Cf. A353481 (characteristic function).
%Y A046388 Cf. A001221, A046404, A056911, A175101.
%Y A046388 Different from A056913, A098905, A225375.
%K A046388 nonn
%O A046388 1,1
%A A046388 _Patrick De Geest_, Jun 15 1998
%E A046388 I removed some ambiguity in the definition and edited the entry, merging in some material from A146166. - _N. J. A. Sloane_, May 09 2013