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A382092 Values taken by gcd(a^2 + b^2 + c^2, a*b*c), where a, b, c are positive integers.

%I A382092 #26 Apr 02 2025 15:04:50
%S A382092 1,2,4,5,8,9,10,13,16,17,18,20,25,26,27,29,32,34,36,37,40,41,45,49,50,
%T A382092 52,53,54,58,61,64,65,68,72,73,74,80,81,82,85,89,90,97,98,100,101,104,
%U A382092 106,108,109,113,116,117,121,122,125,128,130,135,136,137
%N A382092 Values taken by gcd(a^2 + b^2 + c^2, a*b*c), where a, b, c are positive integers.
%C A382092 Numbers k such that for each prime p == 3 (mod 4) dividing k, v_p(k) > 1, where v_p(k) is the p-valuation of k.
%H A382092 Harvard-MIT Math Tournament, <a href="/A382092/a382092.pdf">HMMT February 2025 Team Round Solutions</a>, Problem 10.
%e A382092 3 is not a term because triples (a, b, c) of positive integers such that gcd(a^2 + b^2 + c^2, a*b*c) = 3 do not exist.
%e A382092 9 is a term because gcd(3^2 + 3^2 + 9^2, 3*3*9) = gcd(99, 81) = 9.
%o A382092 (PARI) isok(n) = {
%o A382092     my(f = factor(n));
%o A382092     for(i=1, #f[,1], if(f[i, 1] % 4 == 3 && f[i, 2] <= 1, return(0)));
%o A382092     return(1);
%o A382092 }
%Y A382092 Cf. A002145.
%Y A382092 A001481 and A072437 are subsequences.
%K A382092 nonn,easy
%O A382092 1,2
%A A382092 _Yifan Xie_, Mar 29 2025