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

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, 52, 53, 54, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 108, 109, 113, 116, 117, 121, 122, 125, 128, 130, 135, 136, 137

Offset: 1

Author

Yifan Xie, Mar 29 2025

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
9 is a term because gcd(3^2 + 3^2 + 9^2, 3*3*9) = gcd(99, 81) = 9.
		

Links

• Harvard-MIT Math Tournament, HMMT February 2025 Team Round Solutions, Problem 10.

Crossrefs

Cf. A002145.

A001481 and A072437 are subsequences.

Programs

PARI

isok(n) = {
    my(f = factor(n));
    for(i=1, #f[,1], if(f[i, 1] % 4 == 3 && f[i, 2] <= 1, return(0)));
    return(1);
}