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

1,1

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.

Links

Table of n, a(n) for n=1..57.

Wikipedia, Cantor's pairing function.

Example

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

Prog

(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)))

Crossrefs

Superset of A277557.

Sequence in context: A006613 A204221 A348277 * A014134 A085989 A319879

Adjacent sequences: A337305 A337306 A337307 * A337309 A337310 A337311

Keyword

nonn

Author

Alexander Fraebel, Aug 22 2020

Status

approved