A379815 a(n) is the smallest integer k > n such that sqrt(1/n + 1/k) is a rational number; or 0 if no such k exists.

0, 16, 9, 0, 20, 12, 441, 64, 16, 90, 1089, 36, 4212, 98, 225, 0, 272, 144, 549081, 25, 567, 2156, 13225, 48, 144, 650, 81, 98, 142100, 150, 71622369, 256, 363, 578, 1225, 64, 1332, 684, 468, 360, 41984, 252, 521345889, 198, 180, 559682, 108241, 144, 63, 400, 127449, 117, 1755572, 108, 2420, 392, 4275, 568458

Offset: 1

Author

Felix Huber, Feb 07 2025

Keywords

• nonn

Comments

a(1) = a(4) = a(16) = 0. Proof: See Huber link.

k > n exists for n > 16.

Examples

			a(3) = 9 because sqrt(1/3 + 1/4) = sqrt(7/12) is irrational, sqrt(1/3 + 1/5) = sqrt(8/15) is irrational, sqrt(1/3 + 1/6) = sqrt(1/2) is irrational, sqrt(1/3 + 1/7) = sqrt(10/21) is irrational, sqrt(1/3 + 1/8) = sqrt(11/24) is irrational, but sqrt(1/3 + 1/9) = sqrt(4/9) = 2/3 is rational.
		

Links

• Felix Huber, Proof that a(1) = a(4) = a(16) = 0

Crossrefs

Cf. A002350, A024352, A076600, A378501, A379816.

Cf. A357372, A257522.

Programs

Maple

A379815:=proc(n)
    local k;
    if n=1 or n=4 or n=16 then
        return 0
    else
        for k from n+1 do
            if type(sqrt(1/n+1/k),rational) then
                return k
            fi
        od
    fi;
end proc;
seq(A379815(n),n=1..58);

PARI

a(n) = if ((n==1) || (n==4) || (n==16), return(0)); my(k=n+1); while (!issquare(1/n + 1/k), k++); k; \\ Michel Marcus, Feb 08 2025

Formula

a(n) <= n*A002350(n)^2 - n if n is not a square; a(m^2) <= A076600(m)^2. - Jinyuan Wang, Feb 11 2025