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A078636 a(n) = rad(n*(n+1)).

%I A078636 #28 Apr 04 2025 04:18:32
%S A078636 2,6,6,10,30,42,14,6,30,110,66,78,182,210,30,34,102,114,190,210,462,
%T A078636 506,138,30,130,78,42,406,870,930,62,66,1122,1190,210,222,1406,1482,
%U A078636 390,410,1722,1806,946,330,690,2162,282,42,70,510,1326,1378,318,330,770,798
%N A078636 a(n) = rad(n*(n+1)).
%H A078636 Reinhard Zumkeller, <a href="/A078636/b078636.txt">Table of n, a(n) for n = 1..1000</a>
%F A078636 From _Reinhard Zumkeller_, Aug 05 2003: (Start)
%F A078636 a(n) = rad(n*(n+1)) = rad(n)*rad(n+1).
%F A078636 mu(a(n)) = mu(rad(n*(n+1))) = mu(rad(n))*mu(rad(n+1)), where rad=A007947 and mu=A008683. (End)
%F A078636 From _Reinhard Zumkeller_, Apr 10 2008: (Start)
%F A078636 a(A014601(n)) = A139131(A014601(n)).
%F A078636 a(n) = A139131(n) * A014695(n). (End)
%F A078636 From _Amiram Eldar_, Apr 04 2025: (Start)
%F A078636 a(n) = A007947(A002378(n)).
%F A078636 Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 2/(p*(p+1))) = 0.4716806... (A307868). (End)
%e A078636 a(3) = 6 as rad(3*4) = rad(12) = rad(2*2*3) = 2*3 = 6.
%p A078636 A078636 := proc(n)
%p A078636     A007947(n)*A007947(n+1) ;
%p A078636 end proc:
%p A078636 seq( A078636(n),n=1..10) ; # _R. J. Mathar_, Mar 15 2023
%t A078636 rad[n_] := Times @@ FactorInteger[n][[All, 1]];
%t A078636 a[n_] := rad[n(n+1)];
%t A078636 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 27 2024 *)
%o A078636 (PARI) rad(n)=local(p,i); p=factor(n)[,1]; prod(i=1,length(p),p[i])
%o A078636 for (k=1,100,print1(rad(k*(k+1))", "))
%Y A078636 Cf. A002378, A007947, A008683, A014601, A139131, A307868.
%K A078636 nonn,easy
%O A078636 1,1
%A A078636 _Jon Perry_, Dec 12 2002