A130512 -id:A130512 - cp's OEIS frontend

A130510 ABC conjecture: values of c in the list of "abc-hits".

9, 32, 49, 64, 81, 81, 125, 128, 225, 243, 245, 250, 256, 256, 289, 343, 375, 512, 512, 513, 539, 625, 625, 625, 676, 729, 729, 729, 729, 961, 968, 1025, 1029, 1216, 1331, 1331, 1331, 1369, 1587, 1681, 2048, 2048, 2048, 2057, 2187, 2187, 2187, 2197, 2197

Offset: 1

T. D. Noe, Jun 01 2007

nonn

Let rad(x) be the function that computes the squarefree kernel of x (see A007947). A triple {a,b,c} of positive integers with a+b=c, gcd(a,b)=1 and c > rad(a*b*c) is called an abc-hit. The corresponding values of a and rad(a*b*c) are in the sequences A130511 and A130512.

			81 appears twice because 1+80=81 and 32+49=81 are two abc-hits.

See A120498

T. D. Noe, Table of n, a(n) for n=1..1269 (for c up to 10^6)
Sander R. Dahmen, Lower bounds for numbers of ABC-hits, J. Numb. Theory, Volume 128, Issue 6, June 2008, pp. 1864-1873.
Noam D. Elkies, The ABC's of Number Theory, The Harvard College Mathematics Review, Vol. 1, No. 1, Spring 2007, pp. 57-76.
Brian Hayes, Easy as abc
Wikipedia, abc conjecture

Cf. A120498 (unique values of c).
Cf. A130511, A130512 (a, and rad(a*b*c)).
Cf. A225425 (number of solutions with c < 10^n).
Cf. A225426 (triples of numbers a,b,c).

rad[n_] := If[n==1, 1, Times@@(Transpose[FactorInteger[n]][[1]])]; nn=1000; Do[If[ !PrimeQ[c], Do[b=c-a; If[GCD[a,b]==1 && rad[a*b*c]

from itertools import count, islice
from math import prod, gcd
from sympy import primefactors
def A130510_gen(startvalue=1): # generator of terms >= startvalue
    for c in count(max(startvalue,1)):
        pc = set(primefactors(c))
        for a in range(1,(c>>1)+1):
            b = c-a
            if gcd(a,b)==1 and c>prod(set(primefactors(a))|set(primefactors(b))|pc):
                yield c
A130510_list = list(islice(A130510_gen(),30)) # Chai Wah Wu, Oct 19 2023

A225426 The triples of numbers (a,b,c) that are "abc-hits".

Original entry on oeis.org

1, 8, 9, 5, 27, 32, 1, 48, 49, 1, 63, 64, 1, 80, 81, 32, 49, 81, 4, 121, 125, 3, 125, 128, 1, 224, 225, 1, 242, 243, 2, 243, 245, 7, 243, 250, 13, 243, 256, 81, 175, 256, 1, 288, 289, 100, 243, 343, 32, 343, 375, 5, 507, 512, 169, 343, 512, 1, 512, 513, 27, 512, 539

Offset: 1

T. D. Noe, May 22 2013

nonn

Let rad(x) be the function that computes the squarefree kernel of x (see A007947). A triple {a,b,c} of positive integers with a+b=c, gcd(a,b)=1 and c > rad(a*b*c) is called an abc-hit.

T. D. Noe, Table of (a, b, c)(n), n = 1..1269
Wikipedia, abc conjecture

Cf. A130510, A130511, A130512 (c, a, and rad(a*b*c)).

Cf. A225425 (number of solutions with c < 10^n).

rad[n_] := If[n == 1, 1, Times @@ (Transpose[FactorInteger[n]][[1]])]; nn = 1000; t = {}; r = Table[rad[n], {n, nn}]; Do[If[! PrimeQ[c], Do[b = c - a; If[GCD[a, b] == 1 && r[[a]]*r[[b]]*r[[c]] < c, num++; AppendTo[t, {a, b, c}]], {a, c/2}]], {c, 2, nn}]; t

A216370 Number of ABC triples with quality q > 1 and c < 10^n.

Original entry on oeis.org

1, 6, 31, 120, 418, 1268, 3499, 8987, 22316, 51677, 116978, 252856, 528275, 1075319, 2131671, 4119410, 7801334, 14482059

Offset: 1

Jonathan Vos Post, Sep 05 2012

			a(2) = 6 because there are 6 (a,b,c) triples with c < 10^2 and q > 1. Those triples are {1,8,9}, {1,48,49}, {1,63,64}, {1,80,81}, {5,27,32}, and {32,49,81}.

Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 2004, ISBN 0-387-20860-7.
Carl Pomerance, Computational Number Theory, The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 361-362.

Jordan Ellenberg, Mochizuki on ABC, Sep 03 2012.
Dorian Goldfeld, Beyond the last theorem, Math Horizons, 1996 (September), pp. 26-34.
Reken Mee met ABC, Synthese resultaten, (Dutch), 2011.
Wikipedia, abc conjecture

Cf. A007947, A120498, A130510, A130511, A130512.

rad[n_] := Times @@ Transpose[FactorInteger[n]][[1]]; Table[t = {}; mx = 10^n; Do[c = a + b; If[c < mx && GCD[a, b] == 1 && Log[c] > Log[rad[a*b*c]], AppendTo[t, {a, b, c}]], {a, mx/2}, {b, a, mx - a}]; Length[t], {n, 3}] (* T. D. Noe, Sep 06 2012 *)

A225425 Number of "abc-hits" with c < 10^n.

Original entry on oeis.org

1, 6, 31, 120, 418, 1268, 3499, 8987, 22316, 51677, 116978, 252856, 528275, 1075319, 2131671, 4119410, 7801334, 14482065

Offset: 1

T. D. Noe, May 22 2013

Let rad(x) be the function that computes the squarefree kernel of x (see A007947). A triple {a,b,c} of positive integers with a+b=c, gcd(a,b)=1 and c > rad(a*b*c) is called an abc-hit.

The first 6 terms were computed using A130510. More terms were found on the Wikipedia page.

			The only solution under 10 is 1 + 8 = 9.

Wikipedia, abc conjecture

Cf. A130510, A130511, A130512 (c, a, and rad(a*b*c)).

Cf. A225426 (a,b,c in one sequence).

A272234 Least positive integer c such that (n, c-n, c) is an abc-hit.

Original entry on oeis.org

9, 245, 128, 125, 32, 214375, 250, 9, 2057, 2197, 2187, 5021875, 256, 658503, 85184, 6875, 5120, 148046893, 6144, 19683, 327701, 23882769, 2048, 1830125, 729, 3536405, 32, 50653, 19712, 75926359382399, 19683, 81, 2000033, 793071909, 4131, 313046875, 32805, 2366250327

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(2) = 245 because rad(2*243*245) = 2*3*35 = 210 < 245, hence (2, 243, 245) is an abc-hit and (2, c-2, c) isn't an abc-triple for every c < 245.

Wikipedia, abc conjecture

Cf. A272236 (corresponding values of b).
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 a+1 do
if igcd(a,c)=1 then rc:=rad(c):
if ra*rc

More terms from Jinyuan Wang, Jun 08 2022

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

Original entry on oeis.org

8, 243, 125, 121, 27, 214369, 243, 1, 2048, 2187, 2176, 5021863, 243, 658489, 85169, 6859, 5103, 148046875, 6125, 19663, 327680, 23882747, 2025, 1830101, 704, 3536379, 5, 50625, 19683, 75926359382369, 19652, 49, 2000000, 793071875, 4096, 313046839, 32768, 2366250289

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(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.

Wikipedia, abc conjecture

Cf. A272239 (analog of this sequence taking into account that n - the smallest element of the triple).
Cf. A272234 (corresponding values of c).
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 a+1 do
if igcd(a,c)=1 then rc:=rad(c):
if ra*rc

More terms from Jinyuan Wang, Jun 08 2022

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

Original entry on oeis.org

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

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) = 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.

Wikipedia, abc conjecture

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).

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

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

More terms from Jinyuan Wang, Jun 08 2022

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A130510 ABC conjecture: values of c in the list of "abc-hits".

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A225426 The triples of numbers (a,b,c) that are "abc-hits".

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A216370 Number of ABC triples with quality q > 1 and c < 10^n.

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A225425 Number of "abc-hits" with c < 10^n.

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A272234 Least positive integer c such that (n, c-n, c) is an abc-hit.

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

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

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

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