cp's OEIS Frontend

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.)

%I A172120 #18 May 10 2025 08:59:00
%S A172120 7,11,13,15,19,21,25,35,40,47,61,63,73,79,95,97,107,115,121,133,143,
%T A172120 145,149,151,156,166,167,169,181,184,187,191,203,205,207,211,215,221,
%U A172120 223,227,235,241,255,259,271,273,293,295,301,302,323,329,331,333,355,364
%N 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.)
%C A172120 This sequence is related to the ABC conjecture.
%e A172120 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).
%t A172120 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
%o A172120 (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o A172120 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
%Y A172120 Cf. A007947, A147638, A147306, A147639, A147640, A172121.
%K A172120 nonn
%O A172120 1,1
%A A172120 _Artur Jasinski_, Jan 26 2010
%E A172120 Offset 1 and name corrected by _Michel Marcus_, Aug 04 2019