A015762 - OEIS

%I #31 Sep 08 2022 08:44:40

%S 1,2,3,6,249,498

%N Numbers n such that phi(n) | sigma_4(n).

%C sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

%C sigma_{8j+4}(x)/phi(x) is an integer for j=0..500, x=1,2,3,6,249,498, and this is conjectured to hold for possible larger terms of A015762 and all j. Compare with comments to A015759, A091285, A015770. - _Labos Elemer_, May 27 2004

%C For any odd n in this sequence, 2n is also in the sequence, since phi(2n) = phi(n) and sigma_4(2n) = 17 sigma_4(n). More generally, if gcd(m,n) = 1 and m and n both are in this sequence, then mn is also in the sequence. No odd prime > 3 can be in the sequence, since if p = 2r + 1, then sigma_4(p) = 8r(2r^3 + 4r^2 + 3r + 1) + 2 is divisible by phi(p) = 2r only for r = 1. The term a(5) = 3*83 is the only odd semiprime term with a factor < 10^5. - _M. F. Hasler_, Aug 21 2017

%C a(7) > 3*10^11, if it exists. - _Giovanni Resta_, Aug 23 2017

%t Select[Range[500],Divisible[DivisorSigma[4,#],EulerPhi[#]]&] (* _Harvey P. Dale_, Dec 16 2012 *)

%o (PARI) select( is(n)=sigma(n,4)%eulerphi(n)==0, [1..10^4])  \\ _M. F. Hasler_, Aug 21 2017

%o (Magma) [n: n in [1..1000]| DivisorSigma(4, n) mod EulerPhi(n) eq 0]; // _Vincenzo Librandi_, Aug 22 2017

%Y Cf. A020492, A015759, A015761, A015763, A015764, A015765, A015766, A015767, A015768, A015769, A015770, A015771, A015773, A015774, A094470.

%K nonn

%O 1,2

%A _Robert G. Wilson v_