A097982 - OEIS

%I #22 Dec 04 2020 09:22:47

%S 1,864,2430,7776,27000,55296,69984,82134,215622,432000,497664,629856,

%T 675000,862488,1499136,1749600,2187000,2667168,3449952,3538944,

%U 4287500,4312440,4478976,4563000,5668704,6912000,10800000,13045131,13799808,16875000,18670176,19773000

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

%D J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 749, pp. 95, 319, Ellipses, Paris, 2004.

%H Amiram Eldar, <a href="/A097982/b097982.txt">Table of n, a(n) for n = 1..143</a>

%e For example: 864 is a term since phi(864) = 288, sigma(864) = 2520, 864 = 2^5*3^3, (288+2520)/6^2 = 78.

%t 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 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 (PARI) rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])

%o is(n)=my(t=(eulerphi(n)+sigma(n))/rad(n)^2);denominator(t)==1 && t>1 \\ _Charles R Greathouse IV_, Feb 19 2013

%Y Subsequence of A121850.

%Y Cf. A000010, A000203, A007947.

%K nonn

%O 1,2

%A _Lekraj Beedassy_, Sep 07 2004

%E More terms from _Tanya Khovanova_, Aug 30 2006

%E a(15)-a(29) from _Donovan Johnson_, Feb 05 2010

%E a(1)=1 and a(30)-a(32) added by _Amiram Eldar_, Dec 04 2020