A172121 -id:A172121 - cp's OEIS frontend

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

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

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

cp's OEIS Frontend

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

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cp's OEIS Frontend

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

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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.)