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A167629 The odd composites c such that c=q*g*j*y/2 and q+g=j*y where q,g,j,y are distinct primes.

%I A167629 #20 Feb 06 2025 08:21:47
%S A167629 105,195,231,399,627,897,935,1023,1443,1581,1729,2465,2915,2967,4123,
%T A167629 4301,4623,4715,5487,7055,7685,7881,8099,9717,10707,11339,12099,12995,
%U A167629 14993,16377,16383,17353,17423,19599,20213,20915,23779,24963,25327
%N A167629 The odd composites c such that c=q*g*j*y/2 and q+g=j*y where q,g,j,y are distinct primes.
%H A167629 Karl-Heinz Hofmann, <a href="/A167629/b167629.txt">Table of n, a(n) for n = 1..10000</a>
%e A167629 a(1) =  3 *  7 *  5 = 105 (q=3, g= 7, j=2, y=5)
%e A167629 a(2) = 13 *  3 *  5 = 195 (q=2, g=13, j=3, y=5)
%e A167629 a(3) =  3 * 11 *  7 = 231 (q=3, g=11, j=2, y=7)
%e A167629 a(4) = 19 *  3 *  7 = 329 (q=2, g=19, j=3, y=7)
%e A167629 a(5) =  3 * 19 * 11 = 627 (q=3, g=19, j=2, y=11)
%o A167629 (Python)
%o A167629 from sympy import primerange, primepi
%o A167629 k_upto = 25327
%o A167629 A167629, primeset = set(), set(primelist:= list(primerange(3, int(k_upto**0.5)+1)))
%o A167629 for x in range (primepi(k_upto**(1/3))):
%o A167629     limit, y = k_upto // (a:=primelist[x]), x
%o A167629     while (b:= primelist[(y:=y+1)]) * (c1:=(a * b - 2)) <= limit:
%o A167629         if c1 in primeset : A167629.add(a * b * c1)
%o A167629         if (c2 := b * 2 - a) in primeset : A167629.add(a * b * c2)
%o A167629     y -= 1
%o A167629     while (b:= primelist[(y:=y+1)]) * (c2:=(b * 2 - a)) <= limit:
%o A167629         if c2 in primeset : A167629.add(a * b * c2)
%o A167629 print(A167629:=sorted(A167629)) # _Karl-Heinz Hofmann_, Jan 30 2025
%Y A167629 Cf. A000040, A157931.
%K A167629 nonn
%O A167629 1,1
%A A167629 _Juri-Stepan Gerasimov_, Nov 07 2009
%E A167629 Corrected and extended by _D. S. McNeil_, Dec 10 2009