A379816 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, 4, 12, 0, 25, 18, 448, 16, 12, 100, 1100, 18, 4225, 112, 240, 18, 289, 36, 549100, 25, 588, 2178, 13248, 72, 45, 676, 108, 126, 142129, 180, 71622400, 64, 396, 612, 1260, 48, 1369, 722, 507, 400, 42025, 294, 521345932, 242, 225, 559728, 108288, 72, 112, 100, 127500, 169, 1755625, 162, 2475, 448, 4332, 568516, 16573100, 150

Offset: 1

Author

Felix Huber, Feb 07 2025

Keywords

• nonn

Comments

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

k > n exists for n > 4.

Continues a(61) <= 53872731025, a(62) = 1984, a(63) = 112, a(64) = 72, a(65) = 4225, a(66) = 2178, a(67) <= 159831244588, a(68) = 1156, a(69) = 268272, a(70) = 8820, a(71) <= 859838400, a(72) = 144, a(73) <= 83265625, a(74) = 136900, a(75) = 300, a(76) = 6498, a(77) = 13552, a(78) = 2106, a(79) = 505600, a(80) = 100, a(81) = 108, a(82) = 6724, a(83) = 558092, a(84) = 147, a(85) = 12145225, a(86) = 447458, a(87) = 68208, a(88) = 8712, a(89) = 22250089, a(90) = 100, a(91) = 225450316, a(92)=1150, a(93) = 565068, a(97) <= 3046267249, a(101) = 10201, a(103) <= 5332206050752, a(107) = 99022508. - R. J. Mathar, Feb 21 2025

For nonsquare n, solutions k (not necessarily the smallest ones) are given by k= n*A002350(n)^2 such that 1/n-1/k= (A002349(n)/A002350(n))^2. - R. J. Mathar, Feb 25 2025

For n=p^2, squared odd primes, solutions k (not necessarily the smallest) are constructed by k=p*(p+1)^2/4 such that 1/n -1/k = 1/p^2-1/k = [(p-1)/(p*(p+1))]^2 is a rational square. For other perfect squares n, start from such a solution for a prime factor p|n, n=(p*f)^2, and multiply the 3 terms of both sides of that solution with 1/f^2 to find a solution for n. - R. J. Mathar, Feb 25 2025

Examples

			a(16) = 18 because sqrt(1/16 - 1/17) = sqrt(1/272) is irrational but sqrt(1/16 - 1/18) = sqrt(1/144) = 1/12 is rational.
		

Links

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

Crossrefs

Cf. A024352, A378501, A379815, A002349, A002350.

Programs

Maple

A379816:=proc(n)
    local k;
    if n=1 or n=4 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(A379816(n),n=1..58);

PARI

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

Extensions

Terms a(59-60) from R. J. Mathar, Feb 12 2025