A147306 -id:A147306 - cp's OEIS frontend

A147307 Numbers A of the constrained search for ABC records described in A147306.

1, 1, 1, 1, 1, 1, 5, 19, 41, 125, 23, 1, 1, 1, 95

Offset: 1

Artur Jasinski, Nov 09 2008

The sequences A147305, a(n) and A147307 are steered by searching for records in the ABC conjecture along increasing C confined as described in A147306, the main entry for these three sequences.

Cf. A085152, A085153, A147298-A147307.

a(n)+A147305(n) = A147306(n). gcd(a(n),A147305(n))=1.

Edited by R. J. Mathar, Aug 24 2009

A147305 Numbers B of the constrained search for ABC records described in A147306.

Original entry on oeis.org

5, 11, 17, 23, 35, 47, 49, 125, 343, 361, 625, 2303, 3887, 5831, 279841

Offset: 1

Artur Jasinski, Nov 09 2008

The sequences a(n), A147306 and A147307 are steered by searching for records in the ABC conjecture along increasing C confined as described in A147306, the main entry for these three sequences.

Cf. A085152, A085153, A147298-A147307.

A147307(n)+a(n) = A147306(n). gcd(A147307(n),a(n))=1.

Edited and 25 replaced by 35 - R. J. Mathar, Aug 24 2009

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

A179019 a(n) = (A179017(n)+1)/2.

Original entry on oeis.org

2, 3, 6, 7, 11, 15, 22, 30, 31, 34, 35, 39, 42, 43, 47, 58, 62, 66, 67, 70, 71, 78, 79, 83, 87, 94, 102, 103, 106, 107, 110, 111, 114, 115, 119, 130, 134, 139, 142, 143, 146, 155, 159, 166, 174, 178, 179, 183, 186, 187, 191, 195, 202, 206, 210, 211, 214, 215, 218, 219

Offset: 1

Artur Jasinski, Jun 24 2010

nonn

For numbers a and c, see A172186 and A179017. Numbers b are this sequence.
These numbers c, with distribution a+b=c such that a=(c-1)/2 and b=(c+1)/2, 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).
There exist no numbers or distributions for which L < log(c)/log((c^2-1)c/4). - Artur Jasinski

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A172120, A172121, A147298, A147299, A147300, A147301, A147302, A147306, A147307, A147638, A147639, A147640, A147641, A147642, A147643, A143700, A179017, A172186.

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 + 1)/2]]], {x, 2, 1000}]; aa

a(n) = A179017(n) - A172186(n). - Hugo Pfoertner, Mar 22 2020

A143701 a(n) is the least odd number 2^n - m minimizing A007947(m*(2^n - m)).

Original entry on oeis.org

1, 3, 7, 15, 27, 63, 125, 243, 343, 999, 1805, 3721, 8181, 16335, 32761, 65533, 112847, 190269, 519375, 1046875, 1953125, 4192479, 8385125, 16775019, 24398405, 66976875, 134216625

Offset: 1

Artur Jasinski, Nov 10 2008

a(n) is the smallest odd number such that the product of distinct prime divisors of (2^n)*a(n)*(2^n-a(n)) is the smallest for the range a(n) <= 2^x - a(n) < 2^x.

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

Cf. A007947, A085152, A085153.
Cf. A147298, A147299, A147300, A147302, A147303, A147305, A147306, A147307.
Cf. A147638, A147639, A147640, A147641, A147642, A147643.

aa = {1}; bb = {1}; 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}]; bb (* Artur Jasinski with assistance of M. F. Hasler *)

a(n) = 2^n - A143700(n).

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

cp's OEIS Frontend

A147307 Numbers A of the constrained search for ABC records described in A147306.

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A147305 Numbers B of the constrained search for ABC records described in A147306.

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

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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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A172121 Complement to A172120. Related to the ABC conjecture.

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A179019 a(n) = (A179017(n)+1)/2.

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A143701 a(n) is the least odd number 2^n - m minimizing A007947(m*(2^n - m)).

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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.)