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A272236 Least positive integer b such that (n, b, n+b) is an abc-hit.

%I A272236 #29 Feb 24 2025 13:12:16
%S A272236 8,243,125,121,27,214369,243,1,2048,2187,2176,5021863,243,658489,
%T A272236 85169,6859,5103,148046875,6125,19663,327680,23882747,2025,1830101,
%U A272236 704,3536379,5,50625,19683,75926359382369,19652,49,2000000,793071875,4096,313046839,32768,2366250289
%N A272236 Least positive integer b such that (n, b, n+b) is an abc-hit.
%C A272236 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 A272236 Wikipedia, <a href="http://en.wikipedia.org/wiki/Abc_conjecture">abc conjecture</a>
%e A272236 a(3) = 125 because rad(3*125*128) = 3*5*2 = 31 < 128, hence (3, 125, 128) is an abc-hit and (3, b, b+3) isn't an abc-hit for every b < 125.
%p A272236 rad:=n -> mul(i,i in factorset(n)):
%p A272236 min_c_for_a:=proc(n) local a,b,c,ra,rc;
%p A272236 for a to n do
%p A272236 ra:=rad(a):
%p A272236 for c from a+1 do
%p A272236 if igcd(a,c)=1 then rc:=rad(c):
%p A272236 if ra*rc<c then b:=c-a:
%p A272236 if ra*rc*rad(b)<c then break fi fi fi od:
%p A272236 print([a,b,c]) od end;
%Y A272236 Cf. A272239 (analog of this sequence taking into account that n - the smallest element of the triple).
%Y A272236 Cf. A272234 (corresponding values of c).
%Y A272236 Cf. A120498, A130510 (possible values of c in abc-hits).
%Y A272236 Cf. A225426 (triples of abc-hits).
%Y A272236 Cf. A130512 (radicals of abc-hits).
%Y A272236 Cf. A007947 (radicals).
%K A272236 nonn
%O A272236 1,1
%A A272236 _Vladimir Letsko_, Apr 23 2016
%E A272236 More terms from _Jinyuan Wang_, Jun 08 2022