A379895 -id:A379895 - cp's OEIS frontend

A379983 Numbers k such that there exists a number 1 <= m <= k-1 and at least two different pairs (x,y), 1 <= x < y <= k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2.

385, 425, 432, 450, 504, 585, 616, 630, 665, 693, 728, 770, 792, 800, 810, 850, 864, 900, 910, 935, 952, 1008, 1015, 1040, 1155, 1170, 1197, 1232, 1260, 1275, 1287, 1296, 1320, 1330, 1350, 1360, 1365, 1386, 1456, 1512, 1540, 1547, 1584, 1600, 1620, 1672, 1680

Offset: 1

Jianing Song, Jan 07 2025

nonn

Numbers k = A355812(r) such that A379895(r) < A355813(r).

The smallest k such that there exists a number 1 <= m <= k-1 and at least three different pairs (x,y), 1 <= x < y <= k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2 is k = 1872: we have 1/300^2 - 1/325^2 = 1/468^2 - 1/585^2 = 1/624^2 - 1/1040^2 = 1/720^2 - 1/1872^2. See the Mathematics Stack Exchange link for more examples, and A380150.

			See a-file for examples.

Jinyuan Wang, Table of n, a(n) for n = 1..2000
Mathematics Stack Exchange, Finding multiple ways of representing a number by a difference of inverse squares
Jianing Song, All examples with k <= 1500

Cf. A355812, A355813, A379895, A380150.

is(n) = my(v=[], m2); for(y=1, n-1, for(x=1, y-1, m2=1/(1/x^2-1/y^2+1/n^2); if(m2==m2\1 && issquare(m2), v=concat(v, [m2])); if(#Set(v)<#v, return(1)))); return(0) \\ See also A379895 for its program

More terms from Jinyuan Wang, Jan 08 2025

A379979 Number of pairs (m,k), 1 <= m < k <= N such that there exists 1 <= x < y < k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2, N = A355812(n).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 14, 16, 17, 19, 21, 23, 25, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 57, 60, 62, 64, 65, 67, 69, 71, 75, 76, 78, 82, 84, 86, 89, 91, 93, 95, 97, 99, 101, 105, 108, 110, 112, 116, 118, 119, 121, 123, 129, 131, 133, 137, 139

Offset: 1

Jianing Song, Jan 07 2025

nonn

Partial sums of A379895.

Let S(N) = {1/x^2 - 1/y^2 : 1 <= x < y <= N}, then binomial(N,2) - a(n) is the size of |S(N)|, N = A355812(n). Note that S_N is the number of distinct energy differences within the first N energy levels of a hydrogen atom.

			a(3) = 5 since A355812(3) = 56, and there are 5 such pairs (m,k), 1 <= m < k <= 56:
(m,k) = (7,35): 1/5^2 - 1/7^2 = 1/7^2 - 1/35^2;
(m,k) = (11,55): 1/10^2 - 1/22^2 = 1/11^2 - 1/55^2;
(m,k) = (22,55): 1/10^2 - 1/11^2 = 1/22^2 - 1/55^2;
(m,k) = (8,56): 1/7^2 - 1/14^2 = 1/8^2 - 1/56^2;
(m,k) = (14,56): 1/7^2 - 1/8^2 = 1/14^2 - 1/56^2.
Correspondingly, the set {1/x^2 - 1/y^2 : 1 <= x < y <= 56} is of size binomial(56,2) - 5.

Jianing Song, Table of n, a(n) for n = 1..307 (corresponding to A355812(n) <= 1500)

Cf. A355812, A379895, A355813.

b(n) = my(v=[], m2); for(y=1, n-1, for(x=1, y-1, m2=1/(1/x^2-1/y^2+1/n^2); if(m2==m2\1 && issquare(m2), v=concat(v, [m2])))); #Set(v) \\ #v gives A355813
my(s=0); for(n=1, 1500, if(b(n)>0, s+=b(n); print1(s, ", ")))

cp's OEIS Frontend

A379983 Numbers k such that there exists a number 1 <= m <= k-1 and at least two different pairs (x,y), 1 <= x < y <= k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2.

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