A147638 -id:A147638 - cp's OEIS frontend

A147643 Numbers A associated with the records of the merit function of the ABC conjecture admitting only C which are powers of 23.

7, 17, 162, 2

Offset: 1

Artur Jasinski, Nov 09 2008

If records of the ABC merit function are listed scanning only parameters C of the form 23^x as described in A147642, a(n) is the value of A associated with B=A147641(n) and C=A147642(n).

OEIS Index entries for sequences related to the abc conjecture.

Cf. A085152, A085153, A147298-A147307, A147638-A147642.

a(n) = A147642(n)-A147641(n).

Edited by M. F. Hasler, Jan 16 2015

A143700 a(n) is the least odd number m minimizing A007947(m*(2^n-m)).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 3, 13, 169, 25, 243, 375, 11, 49, 7, 3, 18225, 71875, 4913, 1701, 144027, 1825, 3483, 2197, 9156027, 131989, 1103, 5103, 38525, 458703, 1523, 3483891, 19283525

Offset: 1

Artur Jasinski, Nov 10 2008

Smallest odd number a(n) such that product of distinct prime divisors of (2^n)*a(n)*(2^n - a(n)) is the smallest available for a(n) <= 2^x - a(n) < 2^x.
Product of distinct prime divisors of (2^n)*a(n)*(2^n - a(n)) is also called radical: rad((2^n)*a(n)*(2^n - a(n))).
For numbers 2^n - a(n) see A143701.
For minimal values of rad((2^n)*a(n)*(2^n - a(n))) see A143702.
Related to the abc conjecture. - M. F. Hasler, Nov 13 2008

Cf. A007947, A085152, A085153, A147298-A147307, A147638-A147643.

a = {{1, 1}}; aa = {1}; bb = {}; rr = {}; Do[logmax = 0; k = 2^x; w = Floor[(k - 1)/2]; Do[m = FactorInteger[n (k - n)]; rad = 1; Do[rad = rad m[[s]][[1]], {s, 1, Length[m]}]; log = Log[k]/Log[rad]; If[log > logmax, bmin = k - n; amax = n; logmax = log; r = rad], {n, 1, w, 2}]; Print[{x, amax}]; AppendTo[aa, amax]; AppendTo[bb, bmin]; AppendTo[rr, r]; AppendTo[a, {x, logmax}], {x, 2, 15}]; aa (* Artur Jasinski with assistance of M. F. Hasler *)

A143700(n) = {my(b=1, m=2^n-b); forstep(a=3, 2^(n-1), 2, A007947(a)*A007947(2^n-a)A007947((2^n-a)*b=a)); b; } \\ M. F. Hasler, Nov 13 2008

a(28)-a(33) from M. F. Hasler, Nov 13 2008

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

A147640 The numbers A associated with the search for records in the ABC conjecture constrained as described in A147639.

Original entry on oeis.org

1, 1, 1, 5, 1, 3, 1701

Offset: 1

Artur Jasinski, Nov 09 2008

The standard way to search for records in the ABC conjecture is to run with the C parameter through all the integers A000027. If this search space is diluted by admitting only powers of 2 as in A147639, the sequence of records changes. This sequence here lists the A such that the triples (A=a(n), B=A147638(n), C=A147639(n)) locate records for this search when C is restricted to powers of 2.

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

Definition and comment edited by R. J. Mathar, Aug 28 2009

A143702 a(n) is the minimal values of A007947((2^n)m(2^n-m)).

Original entry on oeis.org

2, 6, 14, 30, 30, 42, 30, 78, 182, 1110, 570, 1830, 6666, 2310, 2534, 5538, 9870, 20010, 141270, 14070, 480090, 155490, 334110, 1794858, 2463270, 2132130, 2349390

Offset: 1

Artur Jasinski, Nov 10 2008

The product of distinct prime divisors of (2^n)*m*(2^n-m) is also called the radical of that number: rad((2^n)*m*(2^n-m)).

For numbers m see A143700.

Cf. A007947, A085152, A085153, A147298-A147307, A147638-A147643.

aa = {1}; bb = {1}; rr = {2}; Do[logmax = 0; k = 2^x; w = Floor[(k - 1)/2]; Do[m = FactorInteger[n (k - n)]; rad = 1; Do[rad = rad m[[s]][[1]], {s, 1, Length[m]}]; log = Log[k]/Log[rad]; If[log > logmax, bmin = k - n; amax = n; logmax = log; r = rad], {n, 1, w, 2}]; Print[{x, amax}]; AppendTo[aa, amax]; AppendTo[bb, bmin]; AppendTo[rr, 2*r]; AppendTo[a, {x, logmax}], {x, 2, 15}]; rr (* Artur Jasinski with assistance of M. F. Hasler *)

Name changed and a(1) added by Jinyuan Wang, Aug 11 2020

A147641 Numbers B in the triples (A,B,C) that set a record in the L-function of the ABC conjecture if the search for C admits only the restricted integer subset of A009967 as described in A147642.

Original entry on oeis.org

16, 512, 12005, 6436341

Offset: 1

Artur Jasinski, Nov 09 2008

If the ABC conjecture is true this sequence is finite.

For numbers A for this case see A147643.

OEIS Index entries for sequences related to the abc conjecture.

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

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.

A179017 Odd numbers c such that c*(c^2 - 1)/4 is squarefree.

Original entry on oeis.org

3, 5, 11, 13, 21, 29, 43, 59, 61, 67, 69, 77, 83, 85, 93, 115, 123, 131, 133, 139, 141, 155, 157, 165, 173, 187, 203, 205, 211, 213, 219, 221, 227, 229, 237, 259, 267, 277, 283, 285, 291, 309, 317, 331, 347, 355, 357, 365, 371, 373, 381, 389, 403, 411, 419, 421

Offset: 1

Artur Jasinski, Jun 24 2010

nonn

Original title was: "Numbers c such that (c^2-1)c is square free and gcd(c-1,c,c+1)=1", but (c^2-1)c is never squarefree for odd c, and gcd(n,n+1) is always = 1. - M. F. Hasler, Nov 03 2013
These numbers c with distribution a+b=c such that a=(c-1)/2 (see A172186) and b=(c+1)/2 (see A179019) have minimal possible values with function L(a,b,c) = log(c)/log(N(a,b,c)) = log(c)/log((c^2-1)c/4).
This function is minimal orbital in hypothesis (a,b,c).
There are no numbers or distributions which have value L less than log(c)/log((c^2-1)*c/4).
Equivalently, odd squarefree numbers c such that (c^2 - 1)/4 is also squarefree. - Jon E. Schoenfield, Feb 13 2023
The asymptotic density of this sequence is Product_{p prime} (1 - 3/p^2) = A206256 = 0.125486980905... (Tsang, 1985). - Amiram Eldar, Feb 26 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Kai-Man Tsang, The distribution of r-tuples of square-free numbers, Mathematika, Vol. 32, No. 2 (1985), pp. 265-275.

Cf. A172120, A172121, A147298, A147299, A147300, A147301, A147302, A147306, A147307, A147638, A147639, A147640, A147641, A147642, A147643, A143700, A172186, A179019, A206256.

aa = {}; Do[If[(GCD[x, (x - 1)/2] == 1) && (GCD[x, (x + 1)/2] == 1) && (GCD[(x - 1)/2, (x + 1)/2] == 1), If[SquareFreeQ[(x^2 - 1) x/4], AppendTo[aa, x]]], {x, 2, 1000}]; aa

forstep(n=3,421,2,issquarefree(n*(n^2-1)/4)&&print1(n",")) \\ M. F. Hasler, Nov 03 2013

is(n)=n%2 && issquarefree(n) && issquarefree(n^2\4) \\ Charles R Greathouse IV, Mar 11 2014

a(n) = 2*A172186(n) + 1. - Bernard Schott, Mar 06 2023

Edited by M. F. Hasler, Nov 03 2013

A172120 Numbers k for which maxima of the function log(k)/log(N(a, k-a, k)) occurs for two or more distinct values of a. (a < k-a, function N(a, k-a, k) is the squarefree kernel of a(k-a)k and gcd(a, k-a, k) = 1.)

Original entry on oeis.org

7, 11, 13, 15, 19, 21, 25, 35, 40, 47, 61, 63, 73, 79, 95, 97, 107, 115, 121, 133, 143, 145, 149, 151, 156, 166, 167, 169, 181, 184, 187, 191, 203, 205, 207, 211, 215, 221, 223, 227, 235, 241, 255, 259, 271, 273, 293, 295, 301, 302, 323, 329, 331, 333, 355, 364

Offset: 1

Artur Jasinski, Jan 26 2010

nonn

This sequence is related to the ABC conjecture.

			a(1)=7 because the maxima of log(7)/log(N(a, 7-a, 7)) occur at two distinct values, a=1 and a=3. In both cases, log(c)/log(N(a,b,c)) is equal to log(7)/log(42).

Cf. A007947, A147638, A147306, A147639, A147640, A172121.

cc = {}; Do[k = x; w = Floor[(k - 1)/2]; logmax = 0; nmax = 0; nmax1 = 0; radmax = 0; logequal = 0; Do[If[(GCD[n, k] == 1) && (GCD[n, k - n] == 1) && (GCD[k, k - n] == 1), m = FactorInteger[k n (k - n)]; rad = 1; Do[rad = rad m[[s]][[1]], {s, 1, Length[m]}]; log = Log[k]/Log[rad]; If[log == logmax, logequal = log; nmax1 = n];If[log > logmax, nmax = n; logmax = log]], {n, 1, w}]; If[logequal == logmax, AppendTo[cc, k]], {x, 3, 100}]; cc

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
isok(n) = {my(lim = if (n%2, n\2, n/2 - 1), v = vector(lim, k, if (gcd([k, n, n-k]) == 1, log(n)/log(rad(k*(n-k)*n)), 0))); if (#v, #select(x->(x==vecmax(v)), v) > 1);} \\ Michel Marcus, Aug 04 2019

Offset 1 and name corrected by Michel Marcus, Aug 04 2019

A172121 Complement to A172120. Related to the ABC conjecture.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 17, 18, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Artur Jasinski, Jan 26 2010

nonn

Numbers k for which the maximum of the function log(k)/log(N(x,k-x,k)) occurs only for a single value of x (x < k-x, function N(x,k-x,k) is the squarefree kernel of x*(k-x)*k and gcd(x,k-x,k)=1).

Equivalent description without the use of the logarithmic function: Let R(n,k) = rad(n*k*(n-k)) if n is prime to k and otherwise +oo. Also let L(n) = [R(n,k) for k = 1..n]. Then m is in this list <=> min(L(m)) occurs exactly once in L(m). (All minima are listed in A147298.) - Peter Luschny, Aug 05 2019

			Using the equivalent description the rows for prime numbers begin:
[2]  [2]
[3]  [6]
[5]  [10, 30]
[7]  [42, 70, 42]
[11] [110, 66, 66, 154, 330]
[13] [78, 286, 390, 78, 130, 546]
[17] [34, 510, 714, 442, 510, 1122, 1190, 102]
[19] [114, 646, 114, 570, 1330, 1482, 798, 418, 570]
2, 3, 5 and 17 are on the list because the minimum in their row is unique, 7, 11, 19 do not occur because the minimum is more than once in the row.

Cf. A007947, A147638, A147306, A147639, A147640, A172120, A147298.

rad := n -> mul(k, k in numtheory:-factorset(n)):
g := (n, k) -> `if`(igcd(n, k) = 1, 1, infinity):
L := n -> [seq(g(n, k)*rad(n*k*(n-k)), k=1..n/2)]:
isA172121 := n -> nops([ListTools:-SearchAll(min(L(n)), L(n))]) = 1:
select(isA172121, [$1..87]); # Peter Luschny, Aug 05 2019

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
isok(n) = {my(v = vector(n\2, k, if (gcd([k, n, n-k]) == 1, rad(k*(n-k)*n), oo))); if (#v, #select(x->(x==vecmin(v)), v) == 1);} \\ Michel Marcus, Aug 06 2019

Offset 1 and name corrected by Michel Marcus, Aug 04 2019

Prepended 2 to the list by Peter Luschny, Aug 06 2019

cp's OEIS Frontend

A147643 Numbers A associated with the records of the merit function of the ABC conjecture admitting only C which are powers of 23.

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A143700 a(n) is the least odd number m minimizing A007947(m*(2^n-m)).

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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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A147640 The numbers A associated with the search for records in the ABC conjecture constrained as described in A147639.

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A143702 a(n) is the minimal values of A007947((2^n)*m*(2^n-m)).

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Mathematica

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A147641 Numbers B in the triples (A,B,C) that set a record in the L-function of the ABC conjecture if the search for C admits only the restricted integer subset of A009967 as described in A147642.

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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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Examples

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A179017 Odd numbers c such that c*(c^2 - 1)/4 is squarefree.

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Mathematica

PARI

PARI

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A172120 Numbers k for which maxima of the function log(k)/log(N(a, k-a, k)) occurs for two or more distinct values of a. (a < k-a, function N(a, k-a, k) is the squarefree kernel of a*(k-a)*k and gcd(a, k-a, k) = 1.)

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A143702 a(n) is the minimal values of A007947((2^n)m(2^n-m)).

A172120 Numbers k for which maxima of the function log(k)/log(N(a, k-a, k)) occurs for two or more distinct values of a. (a < k-a, function N(a, k-a, k) is the squarefree kernel of a(k-a)k and gcd(a, k-a, k) = 1.)