A120498 -id:A120498 - 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

A147306 Numbers C in successive records of the merit function of the ABC conjecture considering only C from A033845.

Original entry on oeis.org

6, 12, 18, 24, 36, 48, 54, 144, 384, 486, 648, 2304, 3888, 5832, 279936

Offset: 1

Artur Jasinski, Nov 09 2008

In a variant of the ABC conjecture (see A120498) we look at triples (A,B,C) restricted to A+B=C, gcd(A,B)=1, and at the merit function L(A,B,C)=log(C)/log(rad(A*B*C)), where rad() is the squarefree kernel A007947, as usual. Watching for records in L() as C runs through the integers generates A147302. In this sequence here, we admit only the C of the sequence A033845, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.
If the ABC conjecture is true this sequence is finite.
The associated numbers B for this case are A147305, the associated A are A147307.

			(A,B,C) = (1,5,6) defines the first record, L=0.5268.. (A,B,C)=(1,11,12) reaches L=0.5931..
(A,B,C) = (1,17,18) reaches L=0.6249.. The first C-number selected from A033845 that does not generate a new record is 72.

Cf. A085152, A085153, A147298-A147307.

Digits := 120 : A007947 := proc(n) local f,p; f := ifactors(n)[2] ; mul( op(1,p),p=f) ; end:
L := proc(A,B,C) local rad; rad := A007947(A*B*C) ; evalf(log(C)/log(rad)) ; end:
isA033845 := proc(n) local f,p; f := ifactors(n)[2] ; for p in f do if not op(1,p) in {2,3} then RETURN(false) ; fi; od: RETURN( (n mod 2 = 0 ) and (n mod 3 = 0 ) ) ; end:
crek := -1 : for C from 3 do if isA033845(C) then for A from 1 to C/2 do B := C-A ; if gcd(A,B) = 1 then l := L(A,B,C) ; if l > crek then print(C) ; crek := l ; fi; fi; od: fi; od: # R. J. Mathar, Aug 24 2009

Edited by R. J. Mathar, Aug 24 2009

A147639 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 2.

Original entry on oeis.org

4, 8, 16, 32, 64, 128, 1048576

Offset: 1

Artur Jasinski, Nov 09 2008

In a variant of the ABC conjecture (see A120498) we look at triples (A,B,C) restricted to A+B=C, gcd(A,B)=1, and at the merit function L(A,B,C)=log(C)/log(rad(A*B*C)), where rad() is the squarefree kernel A007947, as usual. Watching for records in L() as C runs through the integers generates A147302. In this sequence here, we admit only the C of the sequence A000079, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.
If the ABC conjecture is true this sequence is finite.
The associated numbers B for this case are A147638, the associated A are A147640.

			The case C=2 does not create a valid (A,B,C) triple, so C=4 is the first case, which sets a first record L=0.7737 with (A,B,C)=(1,3,4). The next admitted case, C=8, sets a new record L=0.7879 with (A,B,C)=(1,7,8), and so do (A,B,C)=(1,15,16) with L=0.8151. For C=32, we consider the largest L possible for A<B<C, which is (A,B,C)=(5,27,32) with L=1.0189. The value L=0.839 from (A,B,C)=(1,31,32) at the same C is smaller and discarded.

Abderrahmane Nitaj, The ABC conjecture homepage

Cf. A085152, A085153, A147298-A147307, A147638-A147643.

Digits := 120 : A007947 := proc(n) local f, p; f := ifactors(n)[2] ; mul( op(1, p), p=f) ; end:
L := proc(A, B, C) local rad; rad := A007947(A*B*C) ; evalf(log(C)/log(rad)) ; end:
crek := -1 : for x from 2 do C := 2^x ; for A from 1 to C/2 do B := C-A ; if gcd(A, B) = 1 then l := L(A, B, C) ; if l > crek then print(C) ; crek := l ; fi; fi; od: od: # R. J. Mathar, Aug 28 2009

a(2) corrected by R. J. Mathar, Aug 28 2009

A147642 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 23.

Original entry on oeis.org

23, 529, 12167, 6436343

Offset: 1

Artur Jasinski, Nov 09 2008

In a variant of the ABC conjecture (see A120498) we look at triples (A,B,C) restricted to A+B=C, gcd(A,B)=1, and at the merit function L(A,B,C)=log(C)/log(rad(A*B*C)), where rad() is the squarefree kernel A007947, as usual. Watching for records in L() as C runs through the integers generates A147302. In this sequence here, we admit only the C of the form 23^x, see A009967, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.

For associated B for this case see A147641, for associated A see A147643.

			C= 23 is the first candidate (and therefore by definition a record). Scanning the pairs (A,B) for this C we have L-values of L(1,22,23) = 0.5035, L(2,21,23) = 0.456, ... L(6,17,23) = 0.404, L(7,16,23) = 0.542 ,... L(11,12,23) = 0.428. The largest L-value stems from (A=7,B=16) which means the representative triple of the first record is (A,B,C) = (7,16,23).
C= 23^2= 529 is the next candidate. Scanning again all (A,B) values subject to the constraints we achieve L(17,512,529) = 0.941... (Smaller ones like L(81,448,529) = 0.9123... are discarded). Since the L-value for C=529 is larger than the L-value for C=23, the next record is C=529 with representatives (A,B,C)= (17,512,529).
The third candidate is C= 23^3= 12167. This generates a maximum of L(162,12005,12167) = 1.1089... (smaller values like L(17,12150,12167) = 1.0039.. discarded) which is again larger than the maximum of the previous record (which was 0.941..) So the C-value of 12167 is again a record-holder.

OEIS Index entries for sequences related to the abc conjecture.

Cf. A085152, A085153, A147298 - A147307, A147638 - A147643.

A191100 [Squarefree part of (ABC)]/C for A=3, C=A+B, as a function of B, rounded to nearest integer.

Original entry on oeis.org

2, 6, 1, 6, 4, 1, 21, 6, 1, 30, 33, 2, 5, 42, 2, 6, 26, 2, 57, 30, 2, 13, 69, 0, 8, 78, 1, 42, 5, 10, 93, 6, 2, 102, 105, 2, 28, 114, 13, 30, 62, 5, 129, 66, 1, 20, 28, 2, 11, 30, 2, 78, 40, 2, 165, 42, 10, 174, 177, 3, 6, 186, 7, 6, 98, 22, 201, 102, 2, 210, 213, 0, 110, 222, 5, 114

Offset: 1

Darrell Minor, May 25 2011

Given A,B natural numbers, and C=A+B. The ratio [squarefree part of (ABC)]/C (notation: SQP(ABC)/C) can get arbitrarily small, while the unsolved ABC conjecture (i.e., Oesterle-Masser conjecture) is equivalent to the statement that [SQP(ABC)]^n/C has a minimum value if n>1 (because there are conjectured to be finitely many instances of [SQP(ABC)^(1+epsilon)]

			For B=5, we have C=8 so SQP(ABC)=SQP(120)=2*3*5=30, so SQP(ABC)/C=30/8=3.75, which rounds off to 4.
For B=15, we have C=18 so SQP(ABC)=SQP(810)=2*3*5=30, so SQP(ABC)/C=30/18=1.67, which rounds off to 2.

Eric Weisstein's World of Mathematics, abc Conjecture

Cf. A190846, A191093, A120498.

SQP:=func< n | &*[ f[j, 1]: j in [1..#f] ] where f is Factorization(n) >; A191100:=func< n | Round(SQP(a*n*c)/c) where c is a+n where a is 3 >; [ A191100(n): n in [1..80] ]; // Klaus Brockhaus, May 26 2011

rad(n)=my(f=factor(n)[,1]); prod(i=1,#f,f[i])
a(n)=rad(3*n^2+9*n)\/(n+3) \\ Charles R Greathouse IV, Mar 11 2014

from operator import mul
from sympy import primefactors
def rad(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a(n): return int(round(rad(3*n**2 + 9*n)/(n + 3))) # Indranil Ghosh, May 24 2017

A190846 (Squarefree part of (ABC))/C for A=1, C=A+B, as a function of B, rounded to the nearest integer.

Original entry on oeis.org

1, 2, 2, 2, 5, 6, 2, 1, 3, 10, 6, 6, 13, 14, 2, 2, 6, 6, 10, 10, 21, 22, 6, 1, 5, 3, 2, 14, 29, 30, 2, 2, 33, 34, 6, 6, 37, 38, 10, 10, 41, 42, 22, 7, 15, 46, 6, 1, 1, 10, 26, 26, 6, 6, 14, 14, 57, 58, 30, 30, 61, 21, 1, 2, 65, 66, 34, 34, 69, 70, 6, 6, 73, 15, 8, 38, 77, 78, 10, 0, 3, 82, 42

Offset: 1

Darrell Minor, May 25 2011

Given A, B natural numbers, and C=A+B, the ABC conjecture deals with the ratio of the squarefree part of the product A*B*C, divided by C. Here, B plays the role of the OEIS index n.

			For B=14, we have C=15, so SQP(ABC)=SQP(210)=2*3*5*7=210, so SQP(ABC)/C=210/15=14.
For B=19, we have C=20, so SQP(ABC)=SQP(380)=2*5*19=190, so SQP(ABC)/C=190/20=9.5, which rounds to 10.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Abderrahmane Nitaj, The ABC conjecture homepage
Eric Weisstein's World of Mathematics, abc Conjecture
Wikipedia, ABC conjecture

Cf. A191093, A191100, A120498.

SQP:=func< n | &*[ f[j, 1]: j in [1..#f] ] where f is Factorization(n) >; A190846:=func< n | Round(SQP(a*n*c)/c) where c is a+n where a is 1 >; [ A190846(n): n in [1..85] ]; // Klaus Brockhaus, May 27 2011

A190846 := proc(n) c := 1+n ; round(A007947(n*c)/c) ; end proc:
seq(A190846(n),n=1..80) ; # R. J. Mathar, Jun 10 2011

Array[Round[SelectFirst[Reverse@ Divisors[#1 #2], SquareFreeQ]/#2] & @@ {#, # + 1} &, 83] (* Michael De Vlieger, Feb 19 2019 *)

rad(n)=my(f=factor(n)[,1]); prod(i=1,#f,f[i])
a(n)=rad(n^2+n)\/(n+1) \\ Charles R Greathouse IV, Mar 11 2014

from operator import mul
from sympy import primefactors
def rad(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a(n): return int(round(rad(n**2 + n)/(n + 1))) # Indranil Ghosh, May 24 2017

A191093 [Squarefree part of (ABC)]/C for A=2, C=A+B, as a function of B, rounded to nearest integer.

Original entry on oeis.org

2, 1, 6, 1, 10, 1, 5, 1, 6, 3, 22, 3, 26, 1, 30, 0, 34, 2, 38, 5, 42, 3, 9, 3, 1, 7, 6, 7, 58, 1, 62, 1, 66, 3, 70, 3, 74, 5, 78, 5, 82, 11, 29, 11, 30, 3, 13, 1, 14, 3, 102, 1, 106, 1, 110, 7, 114, 15, 118, 15, 41, 1, 42, 1, 130, 17, 134, 17, 138, 3, 142, 3, 29, 19, 30, 19, 154

Offset: 1

Darrell Minor, May 25 2011

			For B=10, we have C=12 so SQP(ABC)=SQP(240)=2*3*5=30, so SQP(ABC)/C=30/12=2.5, which rounds off to 3.
For B=16, we have C=18 so SQP(ABC)=SQP(576)=2*3=6, so SQP(ABC)/C=6/18=0.33, which rounds off to 0.

Wikipedia, abc Conjecture

Cf. A190846, A191100, A120498.

SQP:=func< n | &*[ f[j, 1]: j in [1..#f] ] where f is Factorization(n) >; A191093:=func< n | Round(SQP(a*n*c)/c) where c is a+n where a is 2 >; [ A191093(n): n in [1..80] ]; // Klaus Brockhaus, May 27 2011

rad(n)=my(f=factor(n)[,1]); prod(i=1,#f,f[i])
a(n)=rad(2*n^2+4*n)\/(n+2) \\ Charles R Greathouse IV, Mar 11 2014

from operator import mul
from sympy import primefactors
def rad(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a(n): return int(round(rad(2*n**2 + 4*n)/(n + 2))) # Indranil Ghosh, May 24 2017

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 *)

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

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cp's OEIS Frontend

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

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A147306 Numbers C in successive records of the merit function of the ABC conjecture considering only C from A033845.

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A147639 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 2.

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A147642 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 23.

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A191100 [Squarefree part of (ABC)]/C for A=3, C=A+B, as a function of B, rounded to nearest integer.

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A190846 (Squarefree part of (ABC))/C for A=1, C=A+B, as a function of B, rounded to the nearest integer.

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A191093 [Squarefree part of (ABC)]/C for A=2, C=A+B, as a function of B, rounded to nearest integer.

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