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

1,1

Comments

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.

References

See A120498

Links

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

Example

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

Mathematica

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]<c, Print[{a, b, c, rad[a*b*c]}]], {a, c/2}]], {c, 2, nn}]

Prog

(Python)
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

Crossrefs

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).

Sequence in context: A141573 A075433 A018833 * A120498 A155098 A063134

Adjacent sequences: A130507 A130508 A130509 * A130511 A130512 A130513

Keyword

nonn

Author

T. D. Noe, Jun 01 2007

Status

approved