A379815 -id:A379815 - 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.

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.

Original entry on oeis.org

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

Felix Huber, Feb 07 2025

nonn

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

			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.

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

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

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);

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions