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A172121 Complement to A172120. Related to the ABC conjecture. 3
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, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

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

LINKS

Table of n, a(n) for n=1..72.

EXAMPLE

Using the equivalent description the rows for prime numbers begin:

[2]  [2]

[3]  [6]

[5]  [10, 30]

[7]  [42, 70, 42]

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

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

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

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

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.

MAPLE

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

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

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

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

select(isA172121, [$1..87]); # Peter Luschny, Aug 05 2019

PROG

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

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

CROSSREFS

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

Sequence in context: A112587 A141820 A068090 * A094222 A164043 A172248

Adjacent sequences:  A172118 A172119 A172120 * A172122 A172123 A172124

KEYWORD

nonn

AUTHOR

Artur Jasinski, Jan 26 2010

EXTENSIONS

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

Prepended 2 to the list by Peter Luschny, Aug 06 2019

STATUS

approved

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Last modified August 24 14:11 EDT 2019. Contains 326282 sequences. (Running on oeis4.)