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A016105 Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4).

%I A016105 #69 Sep 26 2024 21:24:05
%S A016105 21,33,57,69,77,93,129,133,141,161,177,201,209,213,217,237,249,253,
%T A016105 301,309,321,329,341,381,393,413,417,437,453,469,473,489,497,501,517,
%U A016105 537,553,573,581,589,597,633,649,669,681,713,717,721,737,749,753,781,789
%N A016105 Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4).
%C A016105 Subsequence of A084109. - _Ralf Stephan_ and _David W. Wilson_, Apr 17 2005
%C A016105 Subsequence of A046388. - _Altug Alkan_, Dec 10 2015
%C A016105 Subsequence of A339817. No common terms with A339870. - _Antti Karttunen_, Dec 26 2020
%C A016105 Named after the Venezuelan-American computer scientist Manuel Blum (b. 1938). - _Amiram Eldar_, Jun 06 2021
%C A016105 First introduced by Blum, Blum, & Shub for the generation of pseudorandom numbers and later applied (by Manuel Blum and other authors) to zero-knowledge proofs. - _Charles R Greathouse IV_, Sep 26 2024
%D A016105 Lenore Blum, Manuel Blum, and Mike Shub. A simple unpredictable pseudorandom number generator, SIAM Journal on computing 15:2 (1986), pp. 364-383.
%H A016105 Antti Karttunen, <a href="/A016105/b016105.txt">Table of n, a(n) for n = 1..26828</a> (all terms < 2^19; first 1000 terms from T. D. Noe)
%H A016105 Joe Hurd, <a href="http://www.gilith.com/research/talks/cambridge1997.pdf">Blum Integers</a>, Talk at the Trinity College, Jan 20 1997.
%H A016105 Wikipedia, <a href="http://en.wikipedia.org/wiki/Blum_integer">Blum integer</a>.
%F A016105 a(n) = A195758(n) * A195759(n). - _Reinhard Zumkeller_, Sep 23 2011
%F A016105 a(n) ~ 4n log n/log log n. - _Charles R Greathouse IV_, Sep 17 2022
%p A016105 N:= 10000: # to get all terms <= N
%p A016105 Primes:= select(isprime, [seq(i,i=3..N/3,4)]):
%p A016105 S:=select(`<=`,{seq(seq(Primes[i]*Primes[j],i=1..j-1),j=2..nops(Primes))},N):
%p A016105 sort(convert(S,list)); # _Robert Israel_, Dec 11 2015
%t A016105 With[{upto = 820}, Select[Union[Times@@@Subsets[ Select[Prime[Range[ PrimePi[ NextPrime[upto/3]]]], Mod[#, 4] == 3 &], {2}]], # <= upto &]] (* _Harvey P. Dale_, Aug 19 2011 *)
%t A016105 Select[4Range[5, 197] + 1, PrimeNu[#] == 2 && MoebiusMu[#] == 1 && Mod[FactorInteger[#][[1, 1]], 4] != 1 &] (* _Alonso del Arte_, Nov 18 2015 *)
%o A016105 (Haskell)
%o A016105 import Data.Set (singleton, fromList, deleteFindMin, union)
%o A016105 a016105 n = a016105_list !! (n-1)
%o A016105 a016105_list = f [3,7] (drop 2 a002145_list) 21 (singleton 21) where
%o A016105    f qs (p:p':ps) t s
%o A016105      | m < t     = m : f qs (p:p':ps) t s'
%o A016105      | otherwise = m : f (p:qs) (p':ps) t' (s' `union` (fromList pqs))
%o A016105      where (m,s') = deleteFindMin s
%o A016105            t' = head $ dropWhile (> 3*p') pqs
%o A016105            pqs = map (p *) qs
%o A016105 -- _Reinhard Zumkeller_, Sep 23 2011
%o A016105 (Perl) use ntheory ":all"; forcomposites { say if ($_ % 4) == 1 && is_square_free($_) && scalar(factor($_)) == 2 && !scalar(grep { ($_ % 4) != 3 } factor($_)); } 10000; # _Dana Jacobsen_, Dec 10 2015
%o A016105 (PARI) list(lim)=my(P=List(),v=List(),t,p); forprimestep(p=3,lim\3,4, listput(P,p)); for(i=2,#P, p=P[i]; for(j=1,i-1, t=p*P[j]; if(t>lim, break); listput(v,t))); Set(v) \\ _Charles R Greathouse IV_, Jul 01 2016, updated Sep 26 2024
%o A016105 (PARI) isA016105(n) = (2==omega(n)&&2==bigomega(n)&&1==(n%4)&&3==((factor(n)[1,1])%4)); \\ _Antti Karttunen_, Dec 26 2020
%o A016105 (Python)
%o A016105 from sympy import factorint
%o A016105 def ok(n):
%o A016105     fn = factorint(n)
%o A016105     return len(fn) == sum(fn.values()) == 2 and all(f%4 == 3 for f in fn)
%o A016105 print([k for k in range(790) if ok(k)]) # _Michael S. Branicky_, Dec 20 2021
%Y A016105 Cf. A002145, A006881, A046388, A339870.
%Y A016105 Intersection of A005117 and A107978.
%Y A016105 Also, subsequence of the following sequences: A046388, A084109, A091113, A167181, A339817.
%K A016105 nonn,easy,nice
%O A016105 1,1
%A A016105 _Robert G. Wilson v_
%E A016105 More terms from _Erich Friedman_