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

1, 864, 2430, 7776, 27000, 55296, 69984, 82134, 215622, 432000, 497664, 629856, 675000, 862488, 1499136, 1749600, 2187000, 2667168, 3449952, 3538944, 4287500, 4312440, 4478976, 4563000, 5668704, 6912000, 10800000, 13045131, 13799808, 16875000, 18670176, 19773000

Offset

1,2

References

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..143

Example

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

Mathematica

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 *)
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 *)

Prog

(PARI) rad(n)=my(f=factor(n)[, 1]); prod(i=1, #f, f[i])
is(n)=my(t=(eulerphi(n)+sigma(n))/rad(n)^2); denominator(t)==1 && t>1 \\ Charles R Greathouse IV, Feb 19 2013

Crossrefs

Subsequence of A121850.

Cf. A000010, A000203, A007947.

Sequence in context: A179671 A297914 A298507 * A298327 A299220 A300034

Adjacent sequences: A097979 A097980 A097981 * A097983 A097984 A097985

Keyword

nonn

Author

Lekraj Beedassy, Sep 07 2004

Extensions

More terms from Tanya Khovanova, Aug 30 2006

a(15)-a(29) from Donovan Johnson, Feb 05 2010

a(1)=1 and a(30)-a(32) added by Amiram Eldar, Dec 04 2020

Status

approved