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

1,3

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.

Links

Table of n, a(n) for n=1..81.

F. Richman, Pythagorean triples.

Wikipedia, Parallelogram.

Wikipedia, Table of divisors.

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))).

Example

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.

Prog

(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*d<n else 0) for d in takewhile(lambda d:d*d<=n, divisors(n))) # Chai Wah Wu, Aug 21 2023

Crossrefs

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

Sequence in context: A309428 A328368 A052297 * A297162 A322587 A372569

Adjacent sequences: A365046 A365047 A365048 * A365050 A365051 A365052

Keyword

nonn

Author

Felix Huber, Aug 18 2023

Status

approved