A147298 Minimum of rad(m (n - m) n) for 0 < m < n, gcd(m,n) = 1, where rad(k) = A007947(k) = product of prime factors of k.

2, 6, 6, 10, 30, 42, 14, 6, 30, 66, 66, 78, 182, 210, 30, 34, 102, 114, 190, 210, 462, 322, 138, 30, 130, 30, 42, 174, 870, 186, 30, 66, 510, 210, 210, 222, 1254, 546, 390, 246, 1722, 258, 946, 330, 690, 1410, 282, 42, 70, 510, 390, 742, 210, 330, 770, 570, 1218

Offset

2,1

Comments

Function rad(k) is used in ABC conjecture applications.

For biggest values of function rad(m n (n - m)) see A147299.

For numbers m for which rad(m n (n - m)) reached minimal value see A147300.

For numbers m for which rad(m n (n - m)) reached maximal value see A147301.

Sequence in each value Log[n]/Log[A147298(n)] reached records see A147297.

Links

Ivan Neretin, Table of n, a(n) for n = 2..1000

Maple

A147298 := proc(n) local rad, g, L;
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)]:
min(L(n)) end: seq(A147298(n), n=2..58); # Peter Luschny, Aug 06 2019

Mathematica

logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; gg \ = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; AppendTo[bb, max]; AppendTo[cc, mmax]; AppendTo[dd, mmin]; AppendTo[gg, ile]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 129}]; aa (*Artur Jasinski*)
Table[Min[Times @@ FactorInteger[#][[All, 1]] & /@ ((m = Select[Range[1, n - 1], GCD[n, #] == 1 &])*(n - m)*n)], {n, 2, 58}] (* Ivan Neretin, May 21 2015 *)

Prog

(PARI) A147298(n)= local(m=n^2); for( a=1, n\2, gcd(a, n)>1 && next; A007947(n-a)*A007947(a)<m || next; m=A007947(n-a)*A007947(a)); m*A007947(n)

Crossrefs

Cf. A007947, A085152, A085153, A147298-A147307.

Sequence in context: A114718 A102261 A245486 * A078636 A083482 A290701

Adjacent sequences: A147295 A147296 A147297 * A147299 A147300 A147301

Keyword

nonn

Author

Artur Jasinski, Nov 05 2008

Status

approved