1,1

C-values are not repeated: (A,B,C)=(13,243,256) and (A,B,C)=(81,175,256) are only represented once, by 256, in the list, for example.

R. J. Mathar and T. D. Noe, Table of n, a(n) for n=1..868

Zenon B. Batang, Squarefree integers and the abc conjecture, arXiv:2109.10226 [math.GM], 2021. See H(c) p. 3; this is the sequence of x such that H(x)>0.

Bart de Smit, Triples of small size [references the ABC@Home project which is inactive since 2015].

A. Granville and T. J. Tucker, It's As Easy As abc, Notices of the AMS, November 2002 (49:10), pp. 1224-1231.

Abderrahmane Nitaj, The ABC Conjecture Home Page.

Ivars Peterson, Math Trek, The Amazing ABC Conjecture [Internet Archive Wayback Machine]

Eric Weisstein's World of Mathematics, abc conjecture

Wikipedia, abc conjecture

OEIS Index entries for sequences related to the abc conjecture.

A+B=C; gcd(A,B)=1; A007947(A*B*C) < C.

For A=1, B=63 and C=64, C=64 is in the list because 1 and 63 are coprime,

because the set of prime factors of 1, 63=3^2*7 and 64=2^6 has the product

of prime factors 3*2*7=42 and this product is smaller than 64.

rad[n_] := Times @@ First /@ FactorInteger[n]; isABC[a_, b_, c_] := (If[a + b != c || GCD[a, b] != 1, Return[0]]; r = rad[a*b*c]; If[r < c, Return[1], Return[0]]); isC[c_] := (For[a = 1, a <= Floor[c/2], a++, If[isABC[a, c - a, c] != 0, Return[1]]]; Return[0]); Select[Range[4000], isC[#] == 1 & ] (* Jean-François Alcover, Jun 24 2013, translated and adapted from Pari *)

(PARI) isABC(a, b, c)={ a+b==c && gcd(a, b)==1 && A007947(a*b*c)<c } \\ Edited by M. F. Hasler, Jan 16 2015

isC(c)={ for(a=1, floor(c/2), if( isABC(a, c-a, c), return(1) )); return(0); }

{ for(n=1, 6000, if( isC(n), print1(n, ", "))) }

(PARI) is_A120498(c)={for(a=1, c\2, gcd(a, c-a)==1 && A007947(a*(c-a)*c)<c && return(1))} \\ M. F. Hasler, Jan 16 2015

(Python)

from itertools import count, islice

from math import prod, gcd

from sympy import primefactors

def A120498_gen(startvalue=1): # generator of terms >= startvalue

for c in count(max(startvalue, 1)):

pc = set(primefactors(c))

for a in range(1, (c>>1)+1):

b = c-a

if gcd(a, b)==1 and c>prod(set(primefactors(a))|set(primefactors(b))|pc):

yield c

break

A120498_list = list(islice(A120498_gen(), 30)) # Chai Wah Wu, Oct 19 2023

Cf. A007947, A085152, A085153.

Cf. A130510 (values of c in the list of "abc-hits").

Sequence in context: A075433 A018833 A130510 * A155098 A063134 A027620

Adjacent sequences: A120495 A120496 A120497 * A120499 A120500 A120501

R. J. Mathar, Aug 06 2006

approved