A379816 -id:A379816 - cp's OEIS frontend

A378501 Integers m such that A379816(m) = A379815(m) + m.

3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87

Offset: 1

Michel Marcus, Feb 10 2025

nonn

Cf. A379815, A379816.

Complement of A380743.

isok(m) = for(k=m+1, 2*m, if(issquare(1/m-1/k), return(0))); m>4||m==3; \\ Jinyuan Wang, Feb 11 2025

More terms from Jinyuan Wang, Feb 11 2025

A380743 Integers m such that A379816(m) != A379815(m) + m.

Original entry on oeis.org

1, 2, 4, 8, 9, 12, 16, 18, 20, 25, 32, 36, 48, 50, 63, 64, 72, 80, 81, 84, 90, 98, 100, 108, 117, 128, 144, 150, 162, 180, 192, 198, 200, 225, 242, 252, 256, 272, 275, 288, 300, 306, 320, 324, 336, 338, 350, 360, 363, 392, 400, 432, 441, 450, 468, 500, 507, 512, 525, 528, 539

Offset: 1

Michel Marcus, Feb 11 2025

nonn

Cf. A379815, A379816.

Complement of A378501.

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.

Original entry on oeis.org

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

Felix Huber, Feb 07 2025

nonn

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

k > n exists for n > 16.

			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.

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

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

Cf. A357372, A257522.

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);

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

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

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A378501 Integers m such that A379816(m) = A379815(m) + m.

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A380743 Integers m such that A379816(m) != A379815(m) + m.

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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.

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