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%I #18 Dec 10 2016 19:44:46
%S 16,18,20,24,27,28,35,44,45,52,55,60,63,65,68,70,76,77,84,85,91,92,95,
%T 99,100,105,110,115,116,117,119,124,130,132,133,140,143,145,148,153,
%U 154,155,156,161,164,165,170,171,172,175,182,185,187,188,190,195,196,203
%N Numbers n for which 9 is a nontrivial quadratic residue mod n but 9 has only trivial quadratic residues +/- 3 mod each of the proper divisors of n.
%C This sequence is a subsequence of A273543; for m,k > 0, a(m) * k is in A273543. All terms of A273543 are of that form. In other words, these are the primitive terms of A273543.
%H Charles R Greathouse IV, <a href="/A274127/b274127.txt">Table of n, a(n) for n = 1..10000</a>
%e x^2 == 9 (mod 16) has solutions 3, 5, 11 and 13, so x^2 == 9 (mod 16 * k) have these solutions, and possibly more as well. Only solutions to the equation x^2 == 9 (mod n) are shown where no proper divisor of n has nontrivial solutions. So 32 isn't shown as 16 has nontrivial solutions as well and is a proper divisor of 32.
%o (PARI) is(n,{t=1})={if(n<16, return(0)); my(v2=valuation(n, 2), v3=valuation(n, 3), k=n/2^v2/3^v3); if(v2<3 && v3<2, if(v2>1, k>1, !isprimepower(k)), if(t==1, d=divisors(n);sum(i=1, #d-1, is(d[i],0))==0,1))} /* _David A. Corneth_, Jun 11 2016, modified PARI-program in A273543 from _Charles R Greathouse IV_ */
%Y Cf. A273543.
%K nonn
%O 1,1
%A _David A. Corneth_, Jun 10 2016
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