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

8, 243, 125, 121, 27, 214369, 243, 1323, 2048, 2187, 2176, 5021863, 243, 658489, 85169, 6859, 5103, 148046875, 6125, 19663, 327680, 23882747, 2025, 1830101, 704, 3536379, 512, 50625, 19683, 75926359382369, 19652, 49, 2000000, 793071875, 4096, 313046839, 32768

Offset: 1

Author

Vladimir Letsko, Apr 23 2016

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Wikipedia, abc conjecture

Crossrefs

Cf. A272240 (corresponding values of c).

Cf. A272236 (analog of this sequence without assumption that n - the smallest element of the triple).

Cf. A120498, A130510 (possible values of c in abc-hits).

Cf. A225426 (triples of abc-hits).

Cf. A130512 (radicals of abc-hits).

Cf. A007947 (radicals).

Programs

Maple

rad:=n -> mul(i,i in factorset(n)):
min_c_for_a:=proc(n) local a,b,c,ra,rc;
for a to n do
ra:=rad(a):
for c from 2*a+1 do
if igcd(a,c)=1 then rc:=rad(c):
if ra*rc
				

PARI

rad(x, y, z) = my(f=factor(x*y*z)[, 1]~); prod(i=1, #f, f[i])
is_abc_hit(x, y, z) = gcd(x, y)==1 && gcd(x, z)==1 && gcd(y, z)==1 && rad(x, y, z) < z
a(n) = my(b=n+1); while(!is_abc_hit(n, b, n+b), b++); b \\ Felix Fröhlich, May 08 2016

Extensions

More terms from Jinyuan Wang, Jun 08 2022