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

1,1

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

OEIS Index entries for sequences related to the 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
A120498_list = list(islice(A120498_gen(), 30)) # Chai Wah Wu, Oct 19 2023

Crossrefs

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

Keyword

nonn

Author

R. J. Mathar, Aug 06 2006

Status

approved