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A272239 Least positive integer b such that b > n and (n, b, n+b) is an abc-hit.

%I A272239 #35 Jun 09 2022 02:30:03
%S A272239 8,243,125,121,27,214369,243,1323,2048,2187,2176,5021863,243,658489,
%T A272239 85169,6859,5103,148046875,6125,19663,327680,23882747,2025,1830101,
%U A272239 704,3536379,512,50625,19683,75926359382369,19652,49,2000000,793071875,4096,313046839,32768
%N A272239 Least positive integer b such that b > n and (n, b, n+b) is an abc-hit.
%C A272239 An abc-hit is a triple of coprime positive integers a, b, c such that a + b = c and rad(abc) < c, where rad(n) is the largest squarefree number dividing n.
%H A272239 Wikipedia, <a href="http://en.wikipedia.org/wiki/Abc_conjecture">abc conjecture</a>
%e A272239 a(8) = 1323 because rad(8*1323*1331) = 2*21*11 = 462 < 1331, hence (8, 1323, 1331) is an abc-hit and (8, b, b+3) isn't an abc-hit for every b where 8 < b < 1323.
%p A272239 rad:=n -> mul(i,i in factorset(n)):
%p A272239 min_c_for_a:=proc(n) local a,b,c,ra,rc;
%p A272239 for a to n do
%p A272239 ra:=rad(a):
%p A272239 for c from 2*a+1 do
%p A272239 if igcd(a,c)=1 then rc:=rad(c):
%p A272239 if ra*rc<c then b:=c-a:
%p A272239 if ra*rc*rad(b)<c then break fi fi fi od:
%p A272239 print([a,b,c]) od end;
%o A272239 (PARI) rad(x, y, z) = my(f=factor(x*y*z)[, 1]~); prod(i=1, #f, f[i])
%o A272239 is_abc_hit(x, y, z) = gcd(x, y)==1 && gcd(x, z)==1 && gcd(y, z)==1 && rad(x, y, z) < z
%o A272239 a(n) = my(b=n+1); while(!is_abc_hit(n, b, n+b), b++); b \\ _Felix Fröhlich_, May 08 2016
%Y A272239 Cf. A272240 (corresponding values of c).
%Y A272239 Cf. A272236 (analog of this sequence without assumption that n - the smallest element of the triple).
%Y A272239 Cf. A120498, A130510 (possible values of c in abc-hits).
%Y A272239 Cf. A225426 (triples of abc-hits).
%Y A272239 Cf. A130512 (radicals of abc-hits).
%Y A272239 Cf. A007947 (radicals).
%K A272239 nonn
%O A272239 1,1
%A A272239 _Vladimir Letsko_, Apr 23 2016
%E A272239 More terms from _Jinyuan Wang_, Jun 08 2022