A130510 -id:A130510 - cp's OEIS frontend

A272240 Least positive integer c such that (n, c-n, c) is an abc-hit and n is the least number in the triple.

9, 245, 128, 125, 32, 214375, 250, 1331, 2057, 2197, 2187, 5021875, 256, 658503, 85184, 6875, 5120, 148046893, 6144, 19683, 327701, 23882769, 2048, 1830125, 729, 3536405, 539, 50653, 19712, 75926359382399, 19683, 81, 2000033, 793071909, 4131, 313046875, 32805

Offset: 1

Vladimir Letsko, Apr 23 2016

nonn

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.

			a(8) = 1331 because rad(8*1323*1331) = 2*21*11 = 462 < 1331, hence (8, 1323, 1331) is an abc-hit and (8, c-8, c) isn't an abc-hit for every c satisfying unequalities c < 1331 and 8 < c-8.

Wikipedia, abc conjecture

Cf. A272239 (corresponding values of b).
Cf. A272234 (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).

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

More terms from Jinyuan Wang, Jun 08 2022

A272242 a(n) is the least number c such that there are exactly n abc-hits with third member c, or 0 if no such c exists.

Original entry on oeis.org

9, 81, 625, 729, 87808, 14641, 130321, 6561, 65536, 59049, 78125

Offset: 1

Vladimir Letsko, Apr 23 2016

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.

Conjecture: a(n) > 0 for all n. - Jianing Song, Sep 21 2018

			a(2) = 81 because there are exactly 2 abc-hits ((1, 80, 81) and (32, 49, 81)) with third member 81 and count of abc-hits with fixed third member c isn't equal to 2 for every c < 81.

Wikipedia, abc conjecture

Cf. A272243.
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).

A272243 a(n) is the smallest number greater than a(n-1) that is expressible as the sum of two positive integers x + y = a(n), so that (x, y, a(n)) is an abc-hit, in more ways than a(n-1).

Original entry on oeis.org

9, 81, 625, 729, 6561, 15625, 117649, 390625

Offset: 1

Vladimir Letsko, Apr 23 2016

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.

Wikipedia, abc conjecture

Cf. A272242.
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).

A276306 Number of pairs of integers (k, m) with k < m < n such that (k, m, n) is an abc-triple.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 3

Felix Fröhlich, Aug 29 2016

nonn

An abc-triple is a set of three integers (a, b, c) such that a+b = c, gcd(a, b) = 1 and rad(a, b, c) < c, where rad() gives the product of the distinct prime factors of its arguments.
a(n) > 0 for n in A120498.
a(n) gives the number of times n appears in A130510.
a(n) gives the number of i such that A225426(A008585(i)) = n.

			For n = 81: there are 2 abc-triples for c = 81 with a < b < c, namely (32, 49, 81) and (1, 80, 81), so a(81) = 2.

Wikipedia, abc conjecture.

Cf. A120498, A130510, A225426.

rad[a_, b_, c_] := Times @@ FactorInteger[a b c][[All, 1]]; abcTripleQ[a_, b_, c_] := a + b == c && GCD[a, b] == 1 && rad[a, b, c] < c; a[n_] := (For[i = 0; m = 1, m <= n-1, m++, For[k = 1, k <= m-1, k++, If[ abcTripleQ[k, m, n], i++]]]; i); Table[a[n], {n, 3, 89}] (* Jean-François Alcover, Sep 04 2016, partly adapted from PARI *)

rad(x, y, z) = my(f=factor(x*y*z)[, 1]~); prod(i=1, #f, f[i])
is_abc_hit(x, y, z) = z==x+y && gcd(x, y)==1 && rad(x, y, z) < z
a(n) = my(i=0); for(m=1, n-1, for(k=1, m-1, if(is_abc_hit(k, m, n), i++))); i

cp's OEIS Frontend

A272240 Least positive integer c such that (n, c-n, c) is an abc-hit and n is the least number in the triple.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A272242 a(n) is the least number c such that there are exactly n abc-hits with third member c, or 0 if no such c exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A272243 a(n) is the smallest number greater than a(n-1) that is expressible as the sum of two positive integers x + y = a(n), so that (x, y, a(n)) is an abc-hit, in more ways than a(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

A276306 Number of pairs of integers (k, m) with k < m < n such that (k, m, n) is an abc-triple.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI