A216370 -id:A216370 - cp's OEIS frontend

A216323 Values for b in abc-triples with a=1.

8, 48, 63, 80, 224, 242, 288, 512, 624, 675, 728, 960, 1024, 1215, 2303, 2400, 3024, 3887, 3968, 4095, 4374, 5831, 6399, 6560, 6655, 6859, 8575, 9375, 9408, 9800, 10647, 12167, 14336, 15624, 16128, 17576, 21951, 24299, 25920, 28125, 29375, 29791

Offset: 1

Wolfdieter Lang, Sep 24 2012

nonn

For abc-triples see de Smit's link.
(a, b, c=a+b) with positive integers a and b, a <= b, gcd(a,b) = 1 is called an abc-triple if r(a,b,c) < c where r(a,b,c) = rad(a*b*c) with rad = A007947 (radical or squarefree kernel). The quality q of an abc-triple is the real positive number q(a,b,c) = log(c)/log(r(a,b,c)), hence q > 1. See also a comment on A216370.
Here one considers a = 1, c = 1+b for b >= 1.
The radical r(1,a(n),a(n)+1) for these abc-triples is 2*A216324.
The highest quality q of the 258 abc-triples (1, a(n), a(n)+1) with b in the range 1..10^7 appears for the triple (1, 4374, 4375) with b = a(21) and q = 1.567887264 (maple 10 digits).
This sequence is infinite because it contains the infinite subsequence b(k) = 9^k - 1, k>=1.
Alvarez-Salazar et al. prove that k is a term iff k/rad(k) > rad(k+1). - Michel Marcus, Jan 05 2023

Frank M Jackson, Table of n, a(n) for n = 1..1500
Elise Alvarez-Salazar, Alexander J. Barrios, Calvin Henaku, and Summer Soller, On abc triples of the form (1,c-1,c), arXiv:2301.01376 [math.NT], 2023.
Bart de Smit, Triples of small size [references the ABC@Home project which is inactive since 2015]
Wolfdieter Lang, Maple program abc1bN.txt for A216323 for b in the range 1..N .

Cf. A007947, A216324, A216370.

read "abc1bN.txt":  abc1bN(30000); (with the above given maple text file).

rad[n_] := Times @@ Transpose[FactorInteger[n]][[1]]; a = 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, b]], {b, a, mx - a}], {n, 5}]; t (* T. D. Noe, Sep 24 2012 *)
Rad[n_] := Module[{lst = FactorInteger[n]}, Times @@ (First /@ lst)]; lst={};
n = 1; While[Length@lst <= 10^2, If[n/Rad[n]>Rad[n+1], AppendTo[lst, n]]; n++]; lst (* Frank M Jackson, Sep 04 2024 *)

(1, b=a(n), a(n)+1) is an abc-triple (which has quality q > 1) with increasingly ordered b values. See the comment above for abc-triples.

A216328 Values for b in abc-triples with a = 2.

Original entry on oeis.org

243, 70225, 265879, 953125, 1015623, 1071873, 1922373, 6436341, 6739605, 7263025

Offset: 1

Wolfdieter Lang, Sep 28 2012

The listed 10 b-values are the ones for all (2,b,2+b) triples
with b from the range {1, 2, ..., 10^7}. The best quality among these values appears for n=8: (2, 6436341, 6436343), b = 3^10*109, with rad(2*b*(2+b)) = 15042 =2*3*23*109 and q(2,6436341,6436343) = 1.629911684 (maple 10 digits). See Tabl. I of the (not updated) link: The ABC Conjecture Home Page.
See A216323 for the list of increasing b values for abc-triples if a=1. There one finds also a reference and a maple program which can be adapted to a=2 instead of a=1.
This sequence is infinite because it contains the infinite subsequence b(k) = 243^(84k+1), k >= 0. - William Hu, Aug 29 2024

			n:  (a=2, b, c=2+a),    rad(a*b*c), q(a*b*c) (maple 10 digits)
1:  (2, 243, 245),         210,     1.028828797
2:  (2, 70225, 70227),     27030,   1.093563284
3:  (2, 255879, 255881),   252642,  1.001024059
4:  (2, 953125, 953127),   525210,  1.045245231
5:  (2, 1015623, 1015625), 128310,  1.175886268
6:  (2, 1071873, 1071875), 926310,  1.010623492
7:  (2, 1922373, 1922375), 799890,  1.064510569
8:  (2, 6436341, 6436343), 15042,   1.629911684
9:  (2, 6739605, 6739607), 3621030, 1.041135746
10: (2, 7263025, 7263027), 94710,   1.378732296
...
From _Wolfdieter Lang_, Oct 02 2012: (Start)
The prime number decomposition of the ten b-values is
3^5, 5^2*53^2, 3^9*13, 5^6*61, 3^2*7^4*47, 3^5*11*401, 3^8*293, 3^10*109, 3^6*5*43^2, 5^2*7^4*11^2.
The ten c = b+2 numbers have the prime number decomposition
5*7^2, 3^5*17^2, 41*79^2, 3^4*7*41^2, 5^7*13, 5^5*7^3, 5^3*7*13^3, 23^5, 7^5*401, 3^11*41. (End)

The ABC Conjecture Home Page, The top ten good abc-examples.

Cf. A216323, A216370.

See the program given in A216323, adapted to a=2.

(2, b=a(n), 2+a(n)) is an abc-triple (which has quality q > 1) with increasingly ordered b values. See the comment above for abc-triples.

cp's OEIS Frontend

A216323 Values for b in abc-triples with a=1.

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