A120498 - OEIS

%I #43 Oct 19 2023 22:31:34

%S 9,32,49,64,81,125,128,225,243,245,250,256,289,343,375,512,513,539,

%T 625,676,729,961,968,1025,1029,1216,1331,1369,1587,1681,2048,2057,

%U 2187,2197,2304,2312,2401,2500,2673,3025,3072,3125,3136,3211,3481,3584,3773,3888

%N Numbers C from the ABC conjecture.

%C 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.

%H R. J. Mathar and T. D. Noe, <a href="/A120498/b120498.txt">Table of n, a(n) for n=1..868</a>

%H Zenon B. Batang, <a href="https://arxiv.org/abs/2109.10226">Squarefree integers and the abc conjecture</a>, arXiv:2109.10226 [math.GM], 2021. See H(c) p. 3; this is the sequence of x such that H(x)>0.

%H Bart de Smit, <a href="http://www.math.leidenuniv.nl/~desmit/abc/index.php?set=1">Triples of small size</a> [references the ABC@Home project which is inactive since 2015].

%H A. Granville and T. J. Tucker, <a href="http://www.ams.org/notices/200210/fea-granville.pdf">It's As Easy As abc</a>, Notices of the AMS, November 2002 (49:10), pp. 1224-1231.

%H Abderrahmane Nitaj, <a href="https://nitaj.users.lmno.cnrs.fr/abc.html">The ABC Conjecture Home Page</a>.

%H Ivars Peterson, Math Trek, <a href="https://web.archive.org/web/20120120075505/http://www.maa.org/mathland/mathtrek_12_8.html">The Amazing ABC Conjecture</a> [Internet Archive Wayback Machine]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/abcConjecture.html">abc conjecture</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/ABC_conjecture">abc conjecture</a>

%H <a href="/index/Ab#abc">OEIS Index entries for sequences related to the abc conjecture</a>.

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

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

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

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

%t 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 *)

%o (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

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

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

%o (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

%o (Python)

%o from itertools import count, islice

%o from math import prod, gcd

%o from sympy import primefactors

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

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

%o         pc = set(primefactors(c))

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

%o             b = c-a

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

%o                 yield c

%o                 break

%o A120498_list = list(islice(A120498_gen(),30)) # _Chai Wah Wu_, Oct 19 2023

%Y Cf. A007947, A085152, A085153.

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

%K nonn

%O 1,1

%A _R. J. Mathar_, Aug 06 2006