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A097982 Numbers n such that (phi(n) + sigma(n))/(rad(n))^2 is an integer > 1 (phi=A000010, sigma=A000203, rad=A007947).

%I A097982 #22 Dec 04 2020 09:22:47
%S A097982 1,864,2430,7776,27000,55296,69984,82134,215622,432000,497664,629856,
%T A097982 675000,862488,1499136,1749600,2187000,2667168,3449952,3538944,
%U A097982 4287500,4312440,4478976,4563000,5668704,6912000,10800000,13045131,13799808,16875000,18670176,19773000
%N A097982 Numbers n such that (phi(n) + sigma(n))/(rad(n))^2 is an integer > 1 (phi=A000010, sigma=A000203, rad=A007947).
%D A097982 J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 749, pp. 95, 319, Ellipses, Paris, 2004.
%H A097982 Amiram Eldar, <a href="/A097982/b097982.txt">Table of n, a(n) for n = 1..143</a>
%e A097982 For example: 864 is a term since phi(864) = 288, sigma(864) = 2520, 864 = 2^5*3^3, (288+2520)/6^2 = 78.
%t A097982 f[n_] := (DivisorSigma[1, n] + EulerPhi[n])/(Times @@ Transpose[FactorInteger[n]][[1]])^2; Do[ If[IntegerQ[f[n] && f[n] != 1], Print[n]], {n, 1, 1000000}] (* _Tanya Khovanova_, Aug 30 2006 *)
%t A097982 f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p - 1)*p^(e - 1); q[1] = True; q[n_] := IntegerQ[(r = (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/ (Times @@ First /@ f)^2)] && r > 1; Select[Range[10^5], q] (* _Amiram Eldar_, Dec 04 2020 *)
%o A097982 (PARI) rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
%o A097982 is(n)=my(t=(eulerphi(n)+sigma(n))/rad(n)^2);denominator(t)==1 && t>1 \\ _Charles R Greathouse IV_, Feb 19 2013
%Y A097982 Subsequence of A121850.
%Y A097982 Cf. A000010, A000203, A007947.
%K A097982 nonn
%O A097982 1,2
%A A097982 _Lekraj Beedassy_, Sep 07 2004
%E A097982 More terms from _Tanya Khovanova_, Aug 30 2006
%E A097982 a(15)-a(29) from _Donovan Johnson_, Feb 05 2010
%E A097982 a(1)=1 and a(30)-a(32) added by _Amiram Eldar_, Dec 04 2020