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%I #17 Aug 27 2019 04:22:33
%S 3,6,19,22,31,51,55,66,94,139,142,159,166,214,235,283,439,534,559,670,
%T 874,946,1726,2131,2419,3559,4759
%N Numbers prime to 7 that are not represented by x^2+y^2+7*z^2.
%C If it exists, a(28) > 10^7. - _Andrew Howroyd_, Aug 17 2019
%H Irving Kaplansky, <a href="https://doi.org/10.1090/S0025-5718-1995-1265017-2">The first nontrivial genus of positive definite ternary forms</a>, Mathematics of Computation, Vol. 64 (1995): 341-345.
%t max = 10000;
%t Table[Table[Table[x^2 + y^2 + 7 z^2, {z, 0, Sqrt[max - x^2 - y^2]/Sqrt[7] // Ceiling}], {y, 0, Sqrt[max - x^2] // Ceiling}], {x, 0, Sqrt[max] // Ceiling}] // Flatten // Union // Select[#, # <= max&]& // Complement[ Select[ Range[max], CoprimeQ[#, 7]&], #]& // Quiet (* _Jean-François Alcover_, Aug 27 2019 *)
%o (PARI)
%o issumsquare(n)={my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1}
%o ok(m)={if(m%7, for(i=0, sqrtint(m\7), if(issumsquare(m-7*i^2), return(0))); 1, 0)}
%o select(ok, [1..10000]) \\ _Andrew Howroyd_, Aug 17 2019
%Y Cf. A097633.
%K nonn,more
%O 1,1
%A _N. J. A. Sloane_, Mar 26 2019
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