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%I #14 Jan 12 2019 02:31:43
%S 2405,3145,4745,6205,6305,8245,8905,9605,12545,12805,14705,16405,
%T 16745,17945,18241,19045,19345,19805,20213,20605,20905,22945,23545,
%U 25805,26605,26945,28645,29705,30073,33745,35705,35989,36205,36305,37505,38369,38545
%N Numbers of the form p*q*r where p, q, r are distinct primes congruent to 1 mod 4 such that Legendre(p/q) = Legendre(p/r) = Legendre(q/r) = -1.
%C If k is a term, the Pell equation x^2 - k*y^2 = -1 has a solution [Dirichlet, Newman (1977)]. This is only a sufficient condition, there are many other solutions, see A031396.
%H Chai Wah Wu, <a href="/A323271/b323271.txt">Table of n, a(n) for n = 1..10000</a>
%H Morris Newman, <a href="https://www.jstor.org/stable/2319968">A note on an equation related to the Pell equation</a>, The American Mathematical Monthly 84.5 (1977): 365-366.
%o (Python)
%o from sympy.ntheory import legendre_symbol, factorint
%o A323271_list, k = [], 1
%o while len(A323271_list) < 10000:
%o fk, fv = zip(*list(factorint(4*k+1).items()))
%o if sum(fv) == len(fk) == 3 and fk[0] % 4 == fk[1] % 4 == fk[2] % 4 == 1 and legendre_symbol(fk[0],fk[1]) == legendre_symbol(fk[0],fk[2]) == legendre_symbol(fk[1],fk[2]) == -1:
%o A323271_list.append(4*k+1)
%o k += 1 # _Chai Wah Wu_, Jan 11 2019
%Y Cf. A002144, A031396, A322781, A323272.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jan 11 2019
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