A130511 -id:A130511 - 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).

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".

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs