A063903 Numbers k such that ud(k)*phi(k) = sigma(k), ud(k) = A034444.

1, 3, 14, 42, 248, 594, 744, 4064, 7668, 12192, 16775168, 50325504, 4294934528, 12884803584, 68719345664, 206158036992

Offset: 1

Author

Jason Earls, Aug 30 2001

Keywords

• nonn
• more

Comments

From Farideh Firoozbakht, Mar 25 2007: (Start)

(1) If 2^p-1 is prime (a Mersenne prime) then 2^(p-2)*(2^p-1) is in the sequence - the proof is easy. So 2^(A000043-2)*(2^A000043-1) is a subsequence of this sequence.

(2) If k is in the sequence and 3 doesn't divide k then 3*k is in the sequence. Hence if 2^p-1 is a Mersenne prime greater than 3 then 3*2^(p-2)*(2^p-1) is in the sequence.

Statement (2) is a special case of "If gcd(m,k)=1 and m & k are in the sequence then m*k is in the sequence (*)". (*) is correct because the three functions ud, phi & sigma are multiplicative.

There is no further term up to 5.6*10^8. (End)

Crossrefs

Cf. A000043, A000668, A020492.

Programs

PARI

ud(n) = 2^omega(n); for(n=1,10^8, if(ud(n)*eulerphi(n)==sigma(n),print(n)))

Extensions

a(11) from R. J. Mathar, Nov 10 2006

a(12) from Farideh Firoozbakht, Mar 25 2007

a(13)-a(16) from Donovan Johnson, Mar 06 2013