

A120498


Numbers C from the ABC conjecture.


16



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
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OFFSET

1,1


COMMENTS

Cvalues 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
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
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 & ] (* JeanFrançois 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, ca, 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, ca)==1 && A007947(a*(ca)*c)<c && return(1))} \\ M. F. Hasler, Jan 16 2015


CROSSREFS

Cf. A007947, A085152, A085153.
Cf. A130510 (values of c in the list of "abchits").
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



