A328971 Numerator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.

2, 3, 5, 4, 7, 7, 5, 9, 6, 7, 9, 11, 11, 7, 8, 9, 11, 13, 8, 13, 15, 13, 9, 10, 11, 12, 15, 10, 17, 11, 12, 15, 13, 17, 19, 11, 17, 13, 19, 17, 19, 12, 13, 14, 21, 15, 19, 16, 21, 13, 14, 23, 19, 16, 23, 17, 21, 14, 25, 23, 16, 17, 25, 21, 23, 15, 19, 16, 17, 18, 23, 27, 25, 19, 20, 16, 17, 23, 18

Offset

1,1

Comments

a(n) / A328972(n) should contain all reduced fractions between 1 and sqrt(2) + 1 without duplicates.

a(n) is built from the difference between the length of the hypotenuse (A020882) and the difference between the two legs (A120682) of the n-th primitive Pythagorean triangle.

A328972(n) (denominators) is built from the difference between the sum of the length of the legs (A120681) and the hypotenuse of the n-th primitive Pythagorean triangle.

Then both numbers are divided by their GCD to get the reduced fraction.

All primitive Pythagorean triangles are sorted first on hypotenuse, then on long leg.

Links

S. Brunner, Table of n, a(n) for n = 1..10000

S. Brunner, List for n = 0..5000 together with the primitive Pythagorean triangles

Example

For n=2 we need the 2nd primitive Pythagorean triangle:
5,12,13
^  ^    We calculate the difference between the two small numbers: 12-5=7.
      ^ And to get our numerator we subtract 7 from the hypotenuse length: 13-7=6.
^  ^    Then we calculate the sum of the two small numbers: 5+12=17.
      ^ We subtract 13 from this sum to get the denominator: 17-13=4.
This gives us the fraction 6/4, and in reduced form 3/2.

Crossrefs

Denominators: A328972.

Sequence in context: A181095 A276345 A257455 * A127515 A332565 A256996

Adjacent sequences: A328968 A328969 A328970 * A328972 A328973 A328974

Keyword

frac,nonn

Author

S. Brunner, Nov 01 2019

Status

approved