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Natural numbers that yield a coprime pair representing a proper fraction under the inverse of Cantor's pairing function.
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%I #31 Mar 26 2023 06:47:40

%S 8,13,18,19,26,32,33,34,41,43,50,52,53,62,64,72,73,74,75,76,85,89,98,

%T 99,100,101,102,103,114,116,118,128,131,133,134,145,147,149,151,162,

%U 163,164,165,166,167,168,169,182,184,188,200,201,202,203,204,205,206

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

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

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pairing_function#Cantor_pairing_function">Cantor's pairing function</a>.

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

%o (Python) # Edited by _M. F. Hasler_, Mar 25 2023

%o from math import gcd

%o def A337308_first(N):

%o L, b = [], 0

%o f = lambda a: (a + b) * (a + b + 1) // 2 + b

%o while N > 0:

%o b += 1

%o if len(L) > 1:

%o L.sort()

%o while L and L[0] < f(1):

%o yield L.pop(0)

%o N -= 1

%o L.extend(f(a) for a in range(1, b) if gcd(a, b) == 1)

%o print(list(A337308_first(50)))

%Y Superset of A277557.

%K nonn

%O 1,1

%A _Alexander Fraebel_, Aug 22 2020