A130510 -id:A130510 - cp's OEIS frontend

A120498 Numbers C from the ABC conjecture.

9, 32, 49, 64, 81, 125, 128, 225, 243, 245, 250, 256, 289, 343, 375, 512, 513, 539, 625, 676, 729, 961, 968, 1025, 1029, 1216, 1331, 1369, 1587, 1681, 2048, 2057, 2187, 2197, 2304, 2312, 2401, 2500, 2673, 3025, 3072, 3125, 3136, 3211, 3481, 3584, 3773, 3888

Offset: 1

R. J. Mathar, Aug 06 2006

nonn

C-values are not repeated: (A,B,C)=(13,243,256) and (A,B,C)=(81,175,256) are only represented once, by 256, in the list, for example.

			For A=1, B=63 and C=64, C=64 is in the list because 1 and 63 are coprime,
because the set of prime factors of 1, 63=3^2*7 and 64=2^6 has the product
of prime factors 3*2*7=42 and this product is smaller than 64.

R. J. Mathar and T. D. Noe, Table of n, a(n) for n=1..868
Zenon B. Batang, Squarefree integers and the abc conjecture, arXiv:2109.10226 [math.GM], 2021. See H(c) p. 3; this is the sequence of x such that H(x)>0.
Bart de Smit, Triples of small size [references the ABC@Home project which is inactive since 2015].
A. Granville and T. J. Tucker, It's As Easy As abc, Notices of the AMS, November 2002 (49:10), pp. 1224-1231.
Abderrahmane Nitaj, The ABC Conjecture Home Page.
Ivars Peterson, Math Trek, The Amazing ABC Conjecture [Internet Archive Wayback Machine]
Eric Weisstein's World of Mathematics, abc conjecture
Wikipedia, abc conjecture
OEIS Index entries for sequences related to the abc conjecture.

Cf. A007947, A085152, A085153.

Cf. A130510 (values of c in the list of "abc-hits").

rad[n_] := Times @@ First /@ FactorInteger[n]; isABC[a_, b_, c_] := (If[a + b != c || GCD[a, b] != 1, Return[0]]; r = rad[a*b*c]; If[r < c, Return[1], Return[0]]); isC[c_] := (For[a = 1, a <= Floor[c/2], a++, If[isABC[a, c - a, c] != 0, Return[1]]]; Return[0]); Select[Range[4000], isC[#] == 1 & ] (* Jean-François Alcover, Jun 24 2013, translated and adapted from Pari *)

isABC(a,b,c)={ a+b==c && gcd(a,b)==1 && A007947(a*b*c)M. F. Hasler, Jan 16 2015
isC(c)={ for(a=1, floor(c/2), if( isABC(a,c-a,c), return(1) )); return(0); }
{ for(n=1,6000, if( isC(n), print1(n,","))) }

is_A120498(c)={for(a=1,c\2, gcd(a,c-a)==1 && A007947(a*(c-a)*c)M. F. Hasler, Jan 16 2015

from itertools import count, islice
from math import prod, gcd
from sympy import primefactors
def A120498_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
                break
A120498_list = list(islice(A120498_gen(),30)) # Chai Wah Wu, Oct 19 2023

A+B=C; gcd(A,B)=1; A007947(A*B*C) < C.

A130512 ABC conjecture: values of rad(abc) in the list of "abc-hits".

Original entry on oeis.org

6, 30, 42, 42, 30, 42, 110, 30, 210, 66, 210, 210, 78, 210, 102, 210, 210, 390, 182, 114, 462, 390, 210, 510, 390, 546, 330, 390, 690, 930, 770, 410, 210, 570, 462, 330, 1122, 1110, 1518, 1230, 690, 1190, 570, 1122, 1122, 834, 2010, 390, 1794, 1974, 510, 210

Offset: 1

T. D. Noe, Jun 01 2007

nonn

T. D. Noe, Table of n, a(n) for n=1..1269

Cf. A130510.

from itertools import count, islice
from math import prod, gcd
from sympy import primefactors
def A130512_gen(): # generator of terms
    for c in count(1):
        pc = set(primefactors(c))
        for a in range(1,(c>>1)+1):
            b = c-a
            if gcd(a,b)==1 and c>(d:=prod(set(primefactors(a))|set(primefactors(b))|pc)):
                yield d
A130512_list = list(islice(A130512_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

A130511 ABC conjecture: values of a in the list of "abc-hits".

Original entry on oeis.org

1, 5, 1, 1, 1, 32, 4, 3, 1, 1, 2, 7, 13, 81, 1, 100, 32, 5, 169, 1, 27, 1, 49, 81, 1, 1, 25, 104, 200, 1, 343, 1, 5, 1, 8, 81, 243, 640, 256, 81, 23, 25, 243, 9, 11, 139, 512, 10, 81, 1, 125, 1, 25, 192, 1024, 99, 625, 1, 875, 53, 128, 11, 1024, 25, 459, 128, 648, 1, 1, 512, 7, 1

Offset: 1

T. D. Noe, Jun 01 2007

nonn

T. D. Noe, Table of n, a(n) for n=1..1269

Cf. A130510.

from itertools import count, islice
from math import prod, gcd
from sympy import primefactors
def A130511_gen(): # generator of terms
    for c in count(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 a
A130511_list = list(islice(A130511_gen(),30)) # Chai Wah Wu, Oct 19 2023

A366428 Hypotenuse numbers w of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit".

Original entry on oeis.org

25, 41, 65, 125, 145, 289, 337, 377, 425, 625, 677, 841, 845, 1025, 1201, 1625, 1681, 1985, 2125, 2197, 2305, 2873, 3125, 3281, 3425, 3721, 4097, 4225, 4481, 4705, 4825, 4901, 4913, 5329, 6401, 6625, 6725, 6845, 7585, 7813, 7817, 8065, 8177, 9409, 10625, 10985

Offset: 1

Felix Huber, Oct 13 2023

nonn

(a, b, c) is an ABC triple if gcd(a, b) = 1 and a + b = c. ABC triples with c > rad(a*b*c) are called "abc-hits". For primitive Pythagorean triples (u, v, w) it is u^2 + v^2 = w^2 and gcd(u^2, v^2) = 1. (u^2, v^2, w^2) are therefore ABC triples. They are then "abc-hits" if in addition w^2 > rad(u^2*v^2*w^2). If (u, v, w) is a non-primitive Pythagorean triple, (u^2, v^2, w^2) is not an ABC triple.
The corresponding values of min(u, v) and max(u, v) are in the sequences A366674 and A366675.
w of primitive Pythagorean triples (u, v, w) with A007947(u^2*v^2*w^2) < w^2.
Subsequence of intersection of A020882 and sqrt(A130510).

			25 from the primitive Pythagorean triple (7, 24, 25) is in the sequence, because 7^2 + 24^2 = 25^2, gcd(7^2, 24^2) = 1 and 25^2 = 625 > rad(7^2*24^2*25^2) = 7*2*3*5 = 210.

Felix Huber, Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit"
Abderrahmane Nitaj, The ABC Conjecture Home Page
Wikipedia, abc conjecture

Cf. A366674, A366675 (corresponding values of min(u, v) and max(u, v)).

Cf. A020882 (hypotenuses of primitive Pythagorean triangles), A130510 ("abc-hits"), A007947 (squarefree kernel).

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

cp's OEIS Frontend

A120498 Numbers C from the ABC conjecture.

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A130512 ABC conjecture: values of rad(a*b*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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A130511 ABC conjecture: values of a in the list of "abc-hits".

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A366428 Hypotenuse numbers w of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit".

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

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Mathematica

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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A130512 ABC conjecture: values of rad(abc) in the list of "abc-hits".