A365049 a(n) is the number of distinct parallelograms with integer sides and area n, and where at least one height is an integer.

1, 1, 2, 3, 2, 4, 2, 5, 5, 4, 2, 10, 2, 4, 8, 9, 2, 9, 2, 10, 8, 4, 2, 20, 5, 4, 8, 10, 2, 16, 2, 13, 8, 4, 8, 23, 2, 4, 8, 20, 2, 16, 2, 10, 18, 4, 2, 34, 5, 9, 8, 10, 2, 16, 8, 20, 8, 4, 2, 40, 2, 4, 18, 19, 8, 16, 2, 10, 8, 16, 2, 45, 2, 4, 18, 10, 8, 16, 2, 34, 13

Offset: 1

Author

Felix Huber, Aug 18 2023

Keywords

• nonn

Comments

If n is not a square, there are A000005(n)/2 rectangles with A027750(n,i)*A027750(n,j) = n, i < j. If n is a square, there are (A000005(n)-1)/2 rectangles with A027750(n,i)*A027750(n,j) = n, i < j and a square with A027750(n,(A000005(n)+1)/2)^2 = n. From these rectangles and, if present, the square, further parallelograms of equal area and integer sides can be formed. A046079(A027750(n,k)) is the number of possibilities there are for each side of the rectangle or for the side of the square.

Examples

			For area n = 9 there is one rectangle (sides of lengths: 1,9) and a square (3,3) with integer sides. From both, further parallelograms with area n = 9 and integer sides can be formed. Since (9,12,15) and (9,40,41) are the only Pythagorean triples with leg 9, from the rectangle (1,9) exactly the two further parallelograms (1,15) and (1,41) with height 9 can be formed, but no further parallelogram with height 1. Since (3,4,5) is the only Pythagorean triple with leg 3, from the square (3,3) exactly one further parallelogram (3,5) with height 3 can be formed. Therefore for area n = 9 there are a(9) = 5 distinct parallelograms with integer sides.
		

Links

• F. Richman, Pythagorean triples.
• Wikipedia, Parallelogram.
• Wikipedia, Table of divisors.

Crossrefs

Cf. A000005, A027750, A046079, A224931, A214602.

Programs

Python

from math import prod
from itertools import takewhile
from sympy import factorint, divisors
def A365049(n): return sum(1+(prod((e+(p&1)<<1)-1 for p, e in factorint(d).items())>>1)+(prod((e+(p&1)<<1)-1 for p, e in factorint(n//d).items())>>1 if d*dChai Wah Wu, Aug 21 2023

Formula

If n is a square, then a(n) = 1 + A046079(A027750(n, (A000005(n) + 1)/2)) + Sum_{i = 1..(A000005(n) - 1)/2} (1 + A046079(A027750(n,i)) + A046079(n/A027750(n,i)));

otherwise, a(n) = Sum_{i = 1..A000005(n)/2} (1 + A046079(A027750(n,i)) + A046079(n/A027750(n,i))).