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.)

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

1,1

Comments

This sequence is related to the ABC conjecture.

Links

Table of n, a(n) for n=1..56.

Example

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).

Mathematica

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

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A370007 A172247 A374082 * A112090 A373673 A103487

Adjacent sequences: A172117 A172118 A172119 * A172121 A172122 A172123

Keyword

nonn

Author

Artur Jasinski, Jan 26 2010

Extensions

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

Status

approved