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

21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, 501, 517, 537, 553, 573, 581, 589, 597, 633, 649, 669, 681, 713, 717, 721, 737, 749, 753, 781, 789

Offset

1,1

Comments

Subsequence of A084109. - Ralf Stephan and David W. Wilson, Apr 17 2005

Subsequence of A046388. - Altug Alkan, Dec 10 2015

Subsequence of A339817. No common terms with A339870. - Antti Karttunen, Dec 26 2020

Named after the Venezuelan-American computer scientist Manuel Blum (b. 1938). - Amiram Eldar, Jun 06 2021

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

References

Lenore Blum, Manuel Blum, and Mike Shub. A simple unpredictable pseudorandom number generator, SIAM Journal on computing 15:2 (1986), pp. 364-383.

Links

Antti Karttunen, Table of n, a(n) for n = 1..26828 (all terms < 2^19; first 1000 terms from T. D. Noe)

Joe Hurd, Blum Integers, Talk at the Trinity College, Jan 20 1997.

Wikipedia, Blum integer.

Formula

a(n) = A195758(n) * A195759(n). - Reinhard Zumkeller, Sep 23 2011

a(n) ~ 4n log n/log log n. - Charles R Greathouse IV, Sep 17 2022

Maple

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(i, i=3..N/3, 4)]):
S:=select(`<=`, {seq(seq(Primes[i]*Primes[j], i=1..j-1), j=2..nops(Primes))}, N):
sort(convert(S, list)); # Robert Israel, Dec 11 2015

Mathematica

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 *)
Select[4Range[5, 197] + 1, PrimeNu[#] == 2 && MoebiusMu[#] == 1 && Mod[FactorInteger[#][[1, 1]], 4] != 1 &] (* Alonso del Arte, Nov 18 2015 *)

Prog

(Haskell)
import Data.Set (singleton, fromList, deleteFindMin, union)
a016105 n = a016105_list !! (n-1)
a016105_list = f [3, 7] (drop 2 a002145_list) 21 (singleton 21) where
   f qs (p:p':ps) t s
     | m < t     = m : f qs (p:p':ps) t s'
     | otherwise = m : f (p:qs) (p':ps) t' (s' `union` (fromList pqs))
     where (m, s') = deleteFindMin s
           t' = head $ dropWhile (> 3*p') pqs
           pqs = map (p *) qs
-- Reinhard Zumkeller, Sep 23 2011
(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
(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
(PARI) isA016105(n) = (2==omega(n)&&2==bigomega(n)&&1==(n%4)&&3==((factor(n)[1, 1])%4)); \\ Antti Karttunen, Dec 26 2020
(Python)
from sympy import factorint
def ok(n):
    fn = factorint(n)
    return len(fn) == sum(fn.values()) == 2 and all(f%4 == 3 for f in fn)
print([k for k in range(790) if ok(k)]) # Michael S. Branicky, Dec 20 2021

Crossrefs

Cf. A002145, A006881, A046388, A339870.

Intersection of A005117 and A107978.

Also, subsequence of the following sequences: A046388, A084109, A091113, A167181, A339817.

Sequence in context: A280262 A084109 A376543 * A187073 A271101 A191683

Adjacent sequences: A016102 A016103 A016104 * A016106 A016107 A016108

Keyword

nonn,easy,nice

Author

Robert G. Wilson v

Extensions

More terms from Erich Friedman

Status

approved