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

Author

Joseph L. Pe, Jan 11 2002

Keywords

• nonn

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

Examples

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

Links

• Jud McCranie, Table of n, a(n) for n = 1..100 (First 71 terms from Donovan Johnson, a(72)-a(93) from Giovanni Resta).
• Douglas E. Iannucci, On the Equation sigma(n) = n + phi(n), Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.

Crossrefs

Cf. A000010, A002235.

Programs

Mathematica

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

PARI

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

Extensions

More terms from Jason Earls, Jan 14 2002

More terms from Farideh Firoozbakht, Apr 15 2006