OFFSET
1,1
COMMENTS
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.
LINKS
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
FORMULA
A+B=C; gcd(A,B)=1; A007947(A*B*C) < C.
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Aug 06 2006
STATUS
approved