A337308 Natural numbers that yield a coprime pair representing a proper fraction under the inverse of Cantor's pairing function.

8, 13, 18, 19, 26, 32, 33, 34, 41, 43, 50, 52, 53, 62, 64, 72, 73, 74, 75, 76, 85, 89, 98, 99, 100, 101, 102, 103, 114, 116, 118, 128, 131, 133, 134, 145, 147, 149, 151, 162, 163, 164, 165, 166, 167, 168, 169, 182, 184, 188, 200, 201, 202, 203, 204, 205, 206

Offset: 1

Author

Alexander Fraebel, Aug 22 2020

Keywords

• nonn

Comments

Equivalently: The image of the function f(x,y)=(x+y)*(x+y+1)/2+y for x,y coprime and 0 < x < y.

Examples

			The fully reduced proper fraction 2/5 is mapped to 33 by Cantor's pairing function.
		

Links

• Wikipedia, Cantor's pairing function.

Crossrefs

Superset of A277557.

Programs

Python

# Edited by M. F. Hasler, Mar 25 2023
from math import gcd
def A337308_first(N):
    L, b = [], 0
    f = lambda a: (a + b) * (a + b + 1) // 2 + b
    while N > 0:
        b += 1
        if len(L) > 1:
            L.sort()
            while L and L[0] < f(1):
                yield L.pop(0)
                N -= 1
        L.extend(f(a) for a in range(1, b) if gcd(a, b) == 1)
print(list(A337308_first(50)))