A216323 -id:A216323 - cp's OEIS frontend

A216324 One half of the radical (squarefree kernel) of the abc-triples (a=1, b(n) = A216323(n), c(n) = 1 + b(n)).

3, 21, 21, 15, 105, 33, 51, 57, 195, 195, 273, 465, 205, 285, 987, 105, 1155, 897, 651, 1365, 105, 357, 1185, 615, 715, 665, 2345, 4395, 2037, 1155, 3003, 897, 1239, 3255, 2667, 8463, 5691, 7755, 2415, 4305, 11985, 4123

Offset: 1

Wolfdieter Lang, Sep 24 2012

nonn

See a comment on A216323 for the definition of an abc-triple, radical r and quality q which is always > 1 by this definition. There also a link is given.

			2*105 = 2*a(21) = r(1,4374,4375) = 1*6*35 = 210.

Wolfdieter Lang, Maple program for radical of a*b*(a+b) .

Cf. A216323.

read "radabc.txt": [seq(radabc(1,A216324(n)),n=1..42)]/2;
(with the above given link with the 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, rad[b*c]/2]], {b, a, mx - a}], {n, 5}]; t (* T. D. Noe, Sep 24 2012 *)

a(n) = r(1,b(n),b(n)+1) with b(n) = A216323(n), n>=1, and r(a,b,c) is the radical, also known as squarefree kernel, of a*b*c.

A376143 Numbers c such that a + b + c = d are abcd quadruples in the "abcd-conjecture" with a < b < c < d, all |a|, b, c, d are pairwise coprime, the quality q of the quadruple has q > 1, term a = +/- 1 = A376149(n) and term b = A376144(n) (with repetitions and sorted by c then b).

Original entry on oeis.org

2187, 2375, 2401, 11881, 14641, 14641, 15125, 16807, 16807, 19321, 22201, 28561, 28561, 42875, 50625, 72171, 77763, 78003, 83349, 83521, 85169, 101761, 116281, 121801, 127253, 128125, 146689, 152881, 177023, 177147, 177147, 249001, 303601, 369603, 390625, 390625

Offset: 1

Frank M Jackson, Sep 11 2024

nonn

An abcd quadruple is defined as (a, b, c, d) with a+b+c+d = 0, all |a|, |b|, |c|, |d| are pairwise coprime, and radical of a*b*c*d, rad(|a|*|b|*|c|*|d|) < max (|a|, |b|, |c|, |d|).
The quality of an abcd quadruple is q = log(max(|a|,|b|,|c|,|d|))/log(rad(|a|*|b|*|c|*|d|)).
This sequence considers quadruples of the form a = +/- 1 and a+b+c = d with a < b < c < d.
Corresponding numbers for b can be found at A376144 and the sequence indicating whether a is 1 or -1 can be found at A376149.
From David A. Corneth, Sep 17 2024: (Start)
All of a, b, c, d in a valid quadruple are odd. Proof: As a + b + c = d, a + b + c + d = 2d is even. Hence an even number of numbers in {a, b, c, d} is odd and as there is an even number of numbers in {a, b, c, d} consequently an even number of them is even.
If more than 0 of them is even then at least two of them are even and not all of {a, b, c, d} are pairwise coprime. Therefore exactly 0 of them are even i.e. they are all odd. (End)

			a(2) = 2375 because the second occurrence of these abcd quadruples with a = +/- 1 is (-1, 27, 2375, 2401) with c = 2375. As prime factors of the form a+d = b+c, we have 1 + 7^4 = 3^3 + 5^3 * 19.
a(4) = 11881 because the fourth occurrence of these abcd quadruples with a = +/- 1 is (1, 25, 11881, 11907) with c = 11881. As prime factors of the form a+b+c = d, we have 1 + 5^2 + 109^2 = 3^5 * 7^2.
From _David A. Corneth_, Sep 16 2024: (Start)
Suppose we look for terms <= 2500. We could iterate over squarefree numbers <= 2500. One of the numbers we will see is 285.
List its prime factors; {3,5,19}.
Make all pairs of sets of distinct prime factors.
These are {{{3}, {5, 19}}, {{5}, {3, 19}}, {{19}, {3,5}}}. One of the pairs we will check is {{3}, {5, 19}}. For each set in it, list all numbers of the form 3^k <= 2500 and 5^m*19^t <= 2500 with k, m, t >= 1. These are (3, 9, 27, 81, 243, 729, 2187) and (95, 475, 1805, 2375). Check all pairs from the Cartesian product of these numbers. One pair is (27, 2375).
The smallest is a candidate for b, the largest for c. See if either (or both) of candidate values a in {-1, 1} give a quadruple (a, b, c, d) (where d = a + b + c meeting the quality requirements and pairwise coprimeness (and d <= 2500)). For this example we find the quadruple (-1, 27, 2375, 2401). (End)

David A. Corneth, Table of n, a(n) for n = 1..161
C. F. W. Ramaekers, The abc-Conjecture and the n-conjecture, Eindhoven University of technology Nov 12, 2009.
David A. Corneth, quadruples (a, b, c, d) with b + c <= 10^7

Cf. A007947, A216323, A376144, A376149.

Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; lst={}; Do[Do[If[d=b+c+a; AllTrue[{{Abs[a],b},{a,c},{a,d},{b,c},{b,d},{c,d}}, Apply[CoprimeQ]]&&d>Rad[Abs[a]*b*c*d], AppendTo[lst,{a,b,c}]], {c, 1, 3000}, {b, 2, c}], {a, {-1, 1}}]; Last/@SortBy[lst,{#[[2]]&,#[[3]]&}]

a(15) corrected by David A. Corneth, Sep 16 2024

A376144 Positive numbers b such that a + b + c = d are abcd quadruples in the "abcd-conjecture" with a < b < c < d, all |a|, b, c, d are pairwise coprime, the quality q of the quadruple has q > 1, term a = +/- 1 = A376149(n) and term c = A376143(n). Quadruples are sorted by c then b.

Original entry on oeis.org

11, 27, 243, 25, 5041, 9747, 1681, 67, 2875, 361, 2187, 841, 16807, 19683, 29, 50653, 361, 121, 173, 513, 125, 28561, 1369, 78125, 78125, 2197, 2187, 243, 125, 95, 3479, 15625, 279841, 83521, 337, 847, 62083, 137781, 378125, 40817, 484183, 343, 8281, 89167, 15625

Offset: 1

Frank M Jackson, Sep 11 2024

nonn

An abcd quadruple is defined as (a, b, c, d) with a+b+c+d = 0, all |a|, |b|, |c|, |d| are pairwise coprime, and radical of a*b*c*d, rad(|a|*|b|*|c|*|d|) < max (|a|, |b|, |c|, |d|).
The quality of an abcd quadruple is q = log(max(|a|,|b|,|c|,|d|))/log(rad(|a|*|b|*|c|*|d|)).
This sequence considers quadruples of the form a = +/- 1 and a+b+c = d with a < b < c < d.
Corresponding numbers c can be found at A376143 and the sequence indicating whether a is 1 or -1 can be found at A376149.

			a(2) = 27 because the second occurrence of an abcd quadruple with a = +/- 1 is (-1, 27, 2375, 2401) with b = 27. As prime factors of the form a+d = b+c we have 1 + 7^4 = 3^3 + 5^3 * 19.
a(4) = 25 because the fourth occurrence of an abcd quadruple with a = +/- 1 is (1, 25, 11881, 11907) with b = 25. As prime factors of the form a+b+c = d we have 1 + 5^2 + 109^2 = 3^5 * 7^2.

David A. Corneth, Table of n, a(n) for n = 1..161
C. F. W. Ramaekers, The abc-Conjecture and the n-conjecture, Eindhoven University of Technology Nov 12, 2009.

Cf. A007947, A216323, A376143, A376149.

Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; lst={}; Do[Do[If[d=b+c+a; AllTrue[{{Abs[a],b},{Abs[a],c},{Abs[a],d},{b,c},{b,d},{c,d}}, Apply[CoprimeQ]]&&d>Rad[Abs[a]*b*c*d], AppendTo[lst,{a,b,c}]], {c, 3, 3000}, {b, 2, c}], {a, {-1, 1}}]; Part[#,2]&/@SortBy[lst,{#[[2]]&,#[[3]]&}]

More terms from David A. Corneth, Sep 18 2024

A376149 Numbers a = +/- 1 such that a + b + c = d are abcd quadruples in the "abcd-conjecture" with a < b < c < d, all |a|, b, c, d are pairwise coprime, the quality q of the quadruple has q > 1, term b = A376144(n) and term c = A376143(n). Quadruples are sorted by c then b.

Original entry on oeis.org

-1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 1, 1

Offset: 1

Frank M Jackson, Sep 12 2024

sign

An abcd quadruple is defined as (a, b, c, d) with a+b+c+d = 0, all |a|, |b|, |c|, |d| are pairwise coprime, and radical of a*b*c*d, rad(|a|*|b|*|c|*|d|) < max (|a|, |b|, |c|, |d|).
The quality of an abcd quadruple is q = log(max(|a|,|b|,|c|,|d|))/log(rad(|a|*|b|*|c|*|d|)).
This sequence considers quadruples of the form a = +/- 1 and a+b+c = d with a < b < c < d.
Corresponding numbers b can be found at A376144 and corresponding numbers c can be found at A376143.

			a(2) = 27 because the second occurrence of an abcd quadruple with a = +/- 1 is (-1, 27, 2375, 2401) with a = -1. As factors of the form a+d = b+c we have 1 + 7^4 = 3^3 + 5^3 * 19.
a(4) = 25 because the fourth occurrence of an abcd quadruple with a = +/- 1 is (1, 25, 11881, 11907) with a = 1. As factors of the form a+b+c = d we have 1 + 5^2 + 109^2 = 3^5 * 7^2.

David A. Corneth, Table of n, a(n) for n = 1..161
C. F. W. Ramaekers, The abc-Conjecture and the n-conjecture, Eindhoven University of Technology Nov 12, 2009.

Cf. A007947, A216323, A376143, A376144.

Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; lst={}; Do[Do[If[d=b+c+a; AllTrue[{{Abs[a],b},{Abs[a],c},{Abs[a],d},{b,c},{b,d},{c,d}}, Apply[CoprimeQ]]&&d>Rad[Abs[a]*b*c*d], AppendTo[lst,{a,b,c}]], {c, 3, 3000}, {b, 2, c}], {a, {-1, 1}}]; First/@SortBy[lst,{#[[2]]&,#[[3]]&}]

More terms from David A. Corneth, Sep 18 2024

A375019 Positive integers n such that (1, n^2 - 1, n^2) is an abc triple.

Original entry on oeis.org

3, 7, 8, 9, 15, 17, 25, 26, 27, 31, 48, 49, 55, 63, 64, 80, 81, 97, 99, 125, 127, 161, 224, 225, 242, 243, 244, 251, 255, 288, 289, 325, 343, 351, 361, 449, 485, 487, 511, 512, 513, 575, 577, 624, 625, 649, 675, 676, 687, 721, 728, 729, 783, 960, 961, 999

Offset: 1

William Hu, Aug 09 2024

nonn

If a number appears in this sequence, so do all of its powers. This immediately implies that this sequence is infinite.
All the terms A216323(n), A216323(n)+1, and 2*A216323(n)+1 appear in this sequence.
The highest known quality abc triple of this form occurs with n = 49, with quality 1.4557, for the triple (1, 2400, 2401).

			For a(1) = 3: (a,b,c) = (1,8,9) is an abc triple. Reason: rad(1*8*9) = rad(72) = 6. Since 6 < 3^2, 3 is a term of this sequence.

William Hu, Table of n, a(n) for n = 1..1816
Bart de Smit, ABC-triples
Wikipedia, abc conjecture

Cf. A007531, A007947, A216323.

is_a375019(n) = factorback(factorint((n-1)*n*(n+1))[,1]) < n^2 \\ Hugo Pfoertner, Aug 09 2024

"""
Note that this code generates all terms <= n, not the nth term.
This code can be further optimized with an O(n log n) sieve, which we do not write here.
"""
n = 10**5  # replace this number with whatever limit
from sympy import primefactors, prod
def rad(n): return 1 if n < 2 else prod(primefactors(n))
# Function to help determine whether a value is a term.
def is_term(k: int):
    # Calculate rad((k^2-1)*k^2) = rad((k-1)*k*(k+1)).
    rad_abc = rad(k-1) * rad(k) * rad(k+1)
    if k % 2 == 1:
        rad_abc //= 2  # 2 is double-counted as a prime factor. No other multiple-counts are possible.
    return rad_abc < k**2
# The final sequence.
a = list(filter(is_term, range(2, n+1))) # William Hu, Aug 09 2024

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.

A375622 Numbers k such that k gives the maximum quality for abc triples of the form (1, k^n-1, k^n) where n is the sequence index.

Original entry on oeis.org

4375, 49, 361, 7, 9, 19, 129, 7, 28, 3, 243, 19, 625, 26, 9, 3

Offset: 1

Frank M Jackson, Aug 21 2024

An abc triple is defined as (a, b, c) with a + b = c, gcd(a, b) = 1 and radical of a*b*c, rad(a*b*c) < c. The quality of an abc triple is q = log(c)/log(rad(a*b*c)). For each n, a sample of the first 100 abc triples of the form (1, k^n-1, k^n) is compared to find the value of k that gives the abc triple maximum quality. The sample size s = 100 of abc triples appears adequate to identify the maximum quality because the quality term tends rapidly towards the lim sup(q) = 1 as s -> oo.

			a(2) = 49 because from the sample of 100 abc triples of the form (1, k^2-1, k^2) (see A375019) where k takes values 3, 7,..., 49,..., 3362, 3375, when k = 49 = A375019(12), we get maximum quality q = 1.45567... with triple (1, 2400, 2401).

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.

Cf. A007947, A216323, A375019.

Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; Table[(lst={}; k=2; While[Length@lst<100, If[Rad[(k^n-1)*k]

cp's OEIS Frontend

A216324 One half of the radical (squarefree kernel) of the abc-triples (a=1, b(n) = A216323(n), c(n) = 1 + b(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A376143 Numbers c such that a + b + c = d are abcd quadruples in the "abcd-conjecture" with a < b < c < d, all |a|, b, c, d are pairwise coprime, the quality q of the quadruple has q > 1, term a = +/- 1 = A376149(n) and term b = A376144(n) (with repetitions and sorted by c then b).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A376144 Positive numbers b such that a + b + c = d are abcd quadruples in the "abcd-conjecture" with a < b < c < d, all |a|, b, c, d are pairwise coprime, the quality q of the quadruple has q > 1, term a = +/- 1 = A376149(n) and term c = A376143(n). Quadruples are sorted by c then b.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A376149 Numbers a = +/- 1 such that a + b + c = d are abcd quadruples in the "abcd-conjecture" with a < b < c < d, all |a|, b, c, d are pairwise coprime, the quality q of the quadruple has q > 1, term b = A376144(n) and term c = A376143(n). Quadruples are sorted by c then b.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A375019 Positive integers n such that (1, n^2 - 1, n^2) is an abc triple.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A375622 Numbers k such that k gives the maximum quality for abc triples of the form (1, k^n-1, k^n) where n is the sequence index.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica