A068418 Composite numbers k such that k - phi(k) divides sigma(k) - k.

12, 56, 260, 992, 1320, 1976, 2156, 2754, 3696, 5520, 13800, 16256, 19872, 22560, 23688, 25232, 41072, 87000, 89964, 133984, 145888, 366720, 785808, 851760, 1100864, 1235052, 1270208, 1439552, 1470720, 2129400, 2237888, 4729664

Offset: 1

Author

Benoit Cloitre, Mar 03 2002

Keywords

• easy
• nonn

Comments

If 2^p - 1 is prime (a Mersenne prime) then k = 2^p*(2^p - 1) is in the sequence because 3*k - 2*phi(k) = sigma(k) (see Comments at A068414) so sigma(k) - k = 2*(k - phi(k)) hence k - phi(k) divides sigma(k) - k. - Farideh Firoozbakht, Dec 31 2005

Also if 3*2^m - 1 is a prime greater than 5 then k = 15*2^(m+1)*(3*2^m - 1) is in the sequence because 4*k - 3*phi(k) = 4*15*2^(m+1)*(3*2^m - 1) - 3*2^(m+3)*(3*2^m - 2) = 24*(3*2^m)*(2^(m+2) - 1) = sigma(15)*sigma(3*2^m - 1)*sigma(2^(m+1)) = sigma(15*(3*2^m - 1)*2^(m+1)) = sigma(k) hence sigma(k) - k = 3*(k - phi(k)) and k - phi(k) divides sigma(k) - k. - Farideh Firoozbakht, Dec 31 2005

Links

• Donovan Johnson, Table of n, a(n) for n = 1..68 (terms < 10^11)

Crossrefs

A068414 is the subsequence telling when the quotient is 2.

Cf. A068414, A051953, A000203, A000668.

Programs

Mathematica

Do[s=(DivisorSigma[1, n]-n)/(n-EulerPhi[n]); If[ !PrimeQ[n]&&IntegerQ[s], Print[n]], {n, 2, 10000000}]

PARI

for(n=1,300000, if((sigma(n)-n)%(n-eulerphi(n))==isprime(n),print1(n,",")))

Extensions

More terms from Labos Elemer, Apr 02 2002