cp's OEIS Frontend

A172121 Complement to A172120. Related to the ABC conjecture.

%I A172121 #33 Aug 07 2019 04:09:35
%S A172121 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,
%T A172121 34,36,37,38,39,41,42,43,44,45,46,48,49,50,51,52,53,54,55,56,57,58,59,
%U A172121 60,62,64,65,66,67,68,69,70,71,72,74,75,76,77,78,80,81,82,83,84,85,86,87
%N A172121 Complement to A172120. Related to the ABC conjecture.
%C A172121 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).
%C A172121 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
%e A172121 Using the equivalent description the rows for prime numbers begin:
%e A172121 [2]  [2]
%e A172121 [3]  [6]
%e A172121 [5]  [10, 30]
%e A172121 [7]  [42, 70, 42]
%e A172121 [11] [110, 66, 66, 154, 330]
%e A172121 [13] [78, 286, 390, 78, 130, 546]
%e A172121 [17] [34, 510, 714, 442, 510, 1122, 1190, 102]
%e A172121 [19] [114, 646, 114, 570, 1330, 1482, 798, 418, 570]
%e A172121 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.
%p A172121 rad := n -> mul(k, k in numtheory:-factorset(n)):
%p A172121 g := (n, k) -> `if`(igcd(n, k) = 1, 1, infinity):
%p A172121 L := n -> [seq(g(n, k)*rad(n*k*(n-k)), k=1..n/2)]:
%p A172121 isA172121 := n -> nops([ListTools:-SearchAll(min(L(n)), L(n))]) = 1:
%p A172121 select(isA172121, [$1..87]); # _Peter Luschny_, Aug 05 2019
%o A172121 (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o A172121 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
%Y A172121 Cf. A007947, A147638, A147306, A147639, A147640, A172120, A147298.
%K A172121 nonn
%O A172121 1,1
%A A172121 _Artur Jasinski_, Jan 26 2010
%E A172121 Offset 1 and name corrected by _Michel Marcus_, Aug 04 2019
%E A172121 Prepended 2 to the list by _Peter Luschny_, Aug 06 2019