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A016105
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Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4).
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14
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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
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OFFSET
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1,1
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COMMENTS
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Named after the Venezuelan-American computer scientist Manuel Blum (b. 1938). - Amiram Eldar, Jun 06 2021
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LINKS
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Joe Hurd, Blum Integers, Talk at the Trinity College, Jan 20 1997.
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FORMULA
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
(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); forprime(p=2, lim\3, if(p%4==3, 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
(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)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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