A066679 Numbers n such that sigma(n) is congruent to n mod phi(n).

1, 2, 6, 10, 12, 44, 90, 184, 440, 528, 588, 672, 752, 3796, 8928, 9888, 12224, 35640, 37680, 49024, 50976, 89152, 94200, 108192, 146412, 159840, 279864, 1734720, 2554368, 2977920, 12580864, 14239872, 16544880, 28321920, 41362200, 56976480, 60610624

Offset

1,2

Comments

Up to 1.5*10^8 there exist 43 terms of the sequence. - Farideh Firoozbakht, Apr 15 2006

If p=3*2^n-1 is an odd prime then m=2^n*p is in the sequence. Proof: sigma(m)-m=(2^(n+1)-1)*(p+1)-2^n*p=2*(2^(n-1)*(p-1))= 2*phi(m), so sigma(m)=m mod(phi(m)). Hence for n>0, 2^A002235(n)* (3*2^A002235(n)-1) is in the sequence and 2^164987*(3*2^164987-1) is the largest known term of the sequence. - Farideh Firoozbakht, Apr 15 2006

Links

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..93 (terms < 10^13, first 71 terms from Donovan Johnson)

Douglas E. Iannucci, On the Equation sigma(n) = n + phi(n), Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.

Example

sigma(10) = 18 is congruent to 10 mod phi(10) = 4, so 10 is a term of the sequence.

Mathematica

Select[ Range[ 1, 10^5 ], Mod[ DivisorSigma[ 1, # ], EulerPhi[ # ] ] == Mod[ #, EulerPhi[ # ] ] & ]

Prog

(PARI) is(n)=sigma(n)==Mod(n, eulerphi(n)) \\ Charles R Greathouse IV, Feb 19 2013

Crossrefs

Cf. A000010, A002235.

Sequence in context: A277238 A108783 A235989 * A086123 A144031 A190055

Adjacent sequences: A066676 A066677 A066678 * A066680 A066681 A066682

Keyword

nonn

Author

Joseph L. Pe, Jan 11 2002

Extensions

More terms from Jason Earls, Jan 14 2002

More terms from Farideh Firoozbakht, Apr 15 2006

Status

approved