A379895 Number of 1 <= m <= N-1 such that there exists 1 <= x < y <= N-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/N^2, N = A355812(n).

1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 4, 1, 2, 4, 2, 2, 3, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 4, 2, 1, 2, 2, 6, 2, 2, 4, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 1, 2, 5, 1, 2, 6, 4, 2, 4, 1, 2, 2, 7

Offset: 1

Author

Jianing Song, Jan 05 2025

Keywords

• nonn

Comments

Different from A355813 at indices r such that A355812(r) is in A379983.

Let S(N) = {1/x^2 - 1/y^2 : 1 <= x < y <= N}, then N - 1 - a(n) is the size of |S(N) \ S(N-1)|, N = A355812(n). Note that S_N is the number of distinct energy differences within the first N energy levels of a hydrogen atom, and in particular |S(N)| < binomial(N,2) for N >= 35 since a(1) = 1.

Examples

			a(65) = 2 since there are 2 such m for N = A355812(65) = 385:
1/77^2 - 1/385^2 = 1/55^2 - 1/77^2 = 1/70^2 - 1/154^2;
1/154^2 - 1/385^2 = 1/70^2 - 1/77^2.
Note that A355813(65) = 3 because there are two solutions (x,y) corresponding to m = 77.
a(204) = 5 since there are 5 such m for N = A355812(204) = 1015:
1/140^2 - 1/1015^2 = 1/116^2 - 1/203^2;
1/203^2 - 1/1015^2 = 1/116^2 - 1/140^2 = 1/145^2 - 1/203^2;
1/609^2 - 1/1015^2 = 1/525^2 - 1/725^2;
1/700^2 - 1/1015^2 = 1/580^2 - 1/725^2;
1/725^2 - 1/1015^2 = 1/525^2 - 1/609^2 = 1/580^2 - 1/700^2.
Note that A355813(204) = 7 because there are two solutions (x,y) corresponding to m = 203 and to m = 725.
		

Links

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

Crossrefs

Cf. A355812, A355813, A379983.

Partial sums give A379979.

Programs

PARI

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
for(n=1, 1500, if(b(n)>0, print1(b(n), ", ")))