A076373 - OEIS

%I #26 Jan 11 2025 05:20:55

%S 10,44,184,752,3796,12224,49024,12580864,60610624,1091389696,

%T 2936313088,46672718384,58082557696,78857645056,118480915456,

%U 206158168064,292776422368,346109272672,393960181792

%N Solutions to n + 2*phi(n) = sigma(n) where phi is A000010 and sigma is A000203.

%C Is the number of solutions finite? Do solutions to n+k*phi(n)=sigma(n) exist for all values of k? For k=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 the number of solutions I know below 1000000 is {1, 7, 2, 2, 1, 5, 3, 3, 0, 1, 1}. Not more for larger k.

%C If 3*2^n-1 is prime for n>0, then 2^n(3*2^n-1) belongs to the sequence; therefore this sequence is infinite if the sequence of primes of the form 3*2^n-1 (A007505) is infinite. - _Matthew Vandermast_, Jul 31 2004

%C 3796 = 4*13*73 and 60610624 = 64*199*4759 do not belong to the class of numbers mentioned above by Vandermast.

%C a(20) > 10^12. - _Donovan Johnson_, Feb 29 2012

%C a(20) > 10^13. - _Giovanni Resta_, Apr 24 2016

%e n=44, phi(n)=20, sigma(44)=1+2+4+11+22+44=84=44+2*20.

%t ta={{0}};k=2;Do[g=n;If[Equal[n+k*EulerPhi[n], DivisorSigma[1, n]], ta=Append[ta, n];Print[n]], {n, 1, 182000000}];{ta, g}

%o (PARI) is(n)=2*eulerphi(n)==sigma(n)-n \\ _Charles R Greathouse IV_, Feb 19 2013

%Y Cf. A000010, A000203, A076374.

%Y A subsequence of A066679.

%K nonn,more

%O 1,1

%A _Labos Elemer_, Oct 15 2002

%E a(8)-a(9) from _Labos Elemer_, Aug 04 2004

%E a(10)-a(19) from _Donovan Johnson_, Feb 29 2012