OFFSET
1,2
COMMENTS
2*k = sigma(k) + phi(k) if and only if k is 1 or a prime.
If 7*2^j - 1 is prime then m = 2^(j+2)*3*(7*2^j - 1) is in the sequence. Because phi(m) = 2^(j+2)*(7*2^j - 2); sigma(m) = 7*2^(j+2)*(2^(j+3) - 1) so phi(m) + sigma(m) = 2^(j+2)*((7*2^j - 2) + (7*2^(j+3) - 7)) = 2^(j+2)* (63*2^(j+2) - 9) = 3*(2^(j+2)*3*(7*2^j - 1)) = 3*m, hence m is a term of A011251 and consequently m is a term of this sequence. A112729 gives such m's. - Farideh Firoozbakht, Dec 01 2005
Conjecture: For n > 1, a(n) is a Zumkeller number (A083207). Verified for all n in [2,63]. - Ivan N. Ianakiev, Jan 25 2023
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
Zhang Ming-Zhi, typescript submitted to Unsolved Problems section of Monthly, 96-01-10.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..63 (terms < 10^13; first 53 terms from Donovan Johnson)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Q.-X. Jin and M. Tang, The 4-Nicol Numbers Having Five Different Prime Divisors, J. Int. Seq. 14 (2011) # 11.7.2.
Eric Weisstein's World of Mathematics, Prime Number.
EXAMPLE
a(26) = 113315400: sigma = 426535200, phi = 26726400, quotient = 4.
MATHEMATICA
Do[If[Mod[DivisorSigma[1, n]+EulerPhi[n], n]==0, Print[n]], {n, 1, 2*10^7}]
Do[ If[ ! PrimeQ[n] && Mod[ DivisorSigma[1, n] + EulerPhi[n], n] == 0, Print[n] ], {n, 1, 10^8} ]
PROG
(PARI) sp(n)=my(f=factor(n)); n*prod(i=1, #f[, 1], 1-1/f[i, 1]) + prod(i=1, #f[, 1], (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1))
p=2; forprime(q=3, 1e6, for(n=p+1, q-1, if(sp(n)%n==0, print1(n", "))); p=q) \\ Charles R Greathouse IV, Mar 19 2012
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from David W. Wilson
Corrected by Labos Elemer, Feb 12 2004
STATUS
approved