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A172121 Complement to A172120. Related to the ABC conjecture. 3

%I #33 Aug 07 2019 04:09:35

%S 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 34,36,37,38,39,41,42,43,44,45,46,48,49,50,51,52,53,54,55,56,57,58,59,

%U 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 Complement to A172120. Related to the ABC conjecture.

%C 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 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 Using the equivalent description the rows for prime numbers begin:

%e [2] [2]

%e [3] [6]

%e [5] [10, 30]

%e [7] [42, 70, 42]

%e [11] [110, 66, 66, 154, 330]

%e [13] [78, 286, 390, 78, 130, 546]

%e [17] [34, 510, 714, 442, 510, 1122, 1190, 102]

%e [19] [114, 646, 114, 570, 1330, 1482, 798, 418, 570]

%e 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 rad := n -> mul(k, k in numtheory:-factorset(n)):

%p g := (n, k) -> `if`(igcd(n, k) = 1, 1, infinity):

%p L := n -> [seq(g(n, k)*rad(n*k*(n-k)), k=1..n/2)]:

%p isA172121 := n -> nops([ListTools:-SearchAll(min(L(n)), L(n))]) = 1:

%p select(isA172121, [$1..87]); # _Peter Luschny_, Aug 05 2019

%o (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947

%o 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 Cf. A007947, A147638, A147306, A147639, A147640, A172120, A147298.

%K nonn

%O 1,1

%A _Artur Jasinski_, Jan 26 2010

%E Offset 1 and name corrected by _Michel Marcus_, Aug 04 2019

%E Prepended 2 to the list by _Peter Luschny_, Aug 06 2019

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)