A120498 Numbers C from the ABC conjecture.

9, 32, 49, 64, 81, 125, 128, 225, 243, 245, 250, 256, 289, 343, 375, 512, 513, 539, 625, 676, 729, 961, 968, 1025, 1029, 1216, 1331, 1369, 1587, 1681, 2048, 2057, 2187, 2197, 2304, 2312, 2401, 2500, 2673, 3025, 3072, 3125, 3136, 3211, 3481, 3584, 3773, 3888

Offset: 1

Author

R. J. Mathar, Aug 06 2006

Keywords

• nonn

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.

Examples

			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.
		

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
• OEIS Index entries for sequences related to the abc conjecture.

Crossrefs

Cf. A007947, A085152, A085153.

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

Programs

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 *)

PARI

isABC(a,b,c)={ a+b==c && gcd(a,b)==1 && A007947(a*b*c)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)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

Formula

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