login
Numbers n such that sum of EulerPhi and DivisorSum is an integer multiple of the number of divisors.
1

%I #20 Dec 01 2017 20:17:15

%S 1,2,3,4,5,7,11,13,15,17,19,21,23,25,29,30,31,33,35,37,39,41,43,45,47,

%T 48,49,51,53,55,56,57,59,61,65,67,69,70,71,73,77,78,79,81,83,85,87,89,

%U 91,93,95,97,99,101,102,103,105,107,109,110,111,113,115,119,121,123

%N Numbers n such that sum of EulerPhi and DivisorSum is an integer multiple of the number of divisors.

%C Makowski proved that phi(n)+Sigma[n] = nd[n] iff n is a prime (see in Sivaramakrishnan, Chapter I, page 8, Theorem 3).

%D Sivaramakrishnan, R. (1989): Classical Theory of Arithmetical Functions Marcel Dekker, Inc., New York-Basel.

%H G. C. Greubel, <a href="/A055464/b055464.txt">Table of n, a(n) for n = 1..10000</a>

%F Solutions to Phi[x]+Sigma[x] = kd[x] or A000203(n)+A000010(n) = k*A000005(n), where k is integer.

%e It is true for all primes and some composites. n = 99, 6 divisors, Sigma = 156, Phi = 60, 156+60 = 216 = 6*36, k = 36.

%t okQ[n_]:=Divisible[EulerPhi[n]+DivisorSigma[1,n],DivisorSigma[0,n]]

%t Select[Range[125],okQ] (* _Harvey P. Dale_, Mar 06 2011 *)

%o (PARI) isok(n) = !((eulerphi(n) + sigma(n)) % numdiv(n)); \\ _Michel Marcus_, Dec 01 2017

%Y Cf. A000005, A065387.

%K nonn

%O 1,2

%A _Labos Elemer_, Jun 27 2000