A076373 Solutions to n + 2*phi(n) = sigma(n) where phi is A000010 and sigma is A000203.

10, 44, 184, 752, 3796, 12224, 49024, 12580864, 60610624, 1091389696, 2936313088, 46672718384, 58082557696, 78857645056, 118480915456, 206158168064, 292776422368, 346109272672, 393960181792

Offset: 1

Author

Labos Elemer, Oct 15 2002

Keywords

• nonn
• more

Comments

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.

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

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

a(20) > 10^12. - Donovan Johnson, Feb 29 2012

a(20) > 10^13. - Giovanni Resta, Apr 24 2016

Examples

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

Crossrefs

Cf. A000010, A000203, A076374.

A subsequence of A066679.

Programs

Mathematica

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}

PARI

is(n)=2*eulerphi(n)==sigma(n)-n \\ Charles R Greathouse IV, Feb 19 2013

Extensions

a(8)-a(9) from Labos Elemer, Aug 04 2004

a(10)-a(19) from Donovan Johnson, Feb 29 2012