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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.

%I A147298 #15 May 10 2025 08:58:29
%S A147298 2,6,6,10,30,42,14,6,30,66,66,78,182,210,30,34,102,114,190,210,462,
%T A147298 322,138,30,130,30,42,174,870,186,30,66,510,210,210,222,1254,546,390,
%U A147298 246,1722,258,946,330,690,1410,282,42,70,510,390,742,210,330,770,570,1218
%N 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.
%C A147298 Function rad(k) is used in ABC conjecture applications.
%C A147298 For biggest values of function rad(m n (n - m)) see A147299.
%C A147298 For numbers m for which rad(m n (n - m)) reached minimal value see A147300.
%C A147298 For numbers m for which rad(m n (n - m)) reached maximal value see A147301.
%C A147298 Sequence in each value Log[n]/Log[A147298(n)] reached records see A147302.
%H A147298 Ivan Neretin, <a href="/A147298/b147298.txt">Table of n, a(n) for n = 2..1000</a>
%p A147298 A147298 := proc(n) local rad, g, L;
%p A147298 rad := n -> mul(k, k in numtheory:-factorset(n)):
%p A147298 g := (n, k) -> `if`(igcd(n, k) = 1, 1, infinity):
%p A147298 L := n -> [seq(g(n,k)*rad(n*k*(n-k)), k=1..n/2)]:
%p A147298 min(L(n)) end: seq(A147298(n), n=2..58); # _Peter Luschny_, Aug 06 2019
%t A147298 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*)
%t A147298 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 *)
%o A147298 (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)
%Y A147298 Cf. A007947, A085152, A085153, A147298-A147307.
%K A147298 nonn
%O A147298 2,1
%A A147298 _Artur Jasinski_, Nov 05 2008