A103176 Let p = prime(sigma(n)) and q = prime(phi(n)), then p is in the sequence if p-q = 6.

13, 19, 43, 113, 463, 619, 863, 1789, 2273, 2383, 4519, 4789, 4937, 5443, 5507, 5653, 8237, 10459, 13007, 13697, 16063, 16453, 17389, 18313, 18919, 20903, 21193, 21319, 21383, 23567, 24109, 25309, 26267, 27947, 28283, 29573, 30559, 31183, 31517

Offset

1,1

Comments

Conjecture: In all cases sigma(n)-phi(n)=2, i.e., n is prime.

Proof: Suppose n is composite. Then sigma(n) > n + sqrt(n) and phi(n) <= n - sqrt(n) and so prime(sigma(n)) - prime(phi(n)) >= sigma(n) - phi(n) > 2*sqrt(n) > 6 for n > 9. - Charles R Greathouse IV, May 15 2013

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

n=3719, sigma(n)=3720, phi(n)=3718, a(n)=p(sigma(n))=34847.

Mathematica

Do[g=n; a=Prime[u=DivisorSigma[1, n]]; b=Prime[w=EulerPhi[n]]; s=a-b; If[Equal[s, 6], Print[{n, a, b, u, w, u-w}]; ta=Append[ta, a]], {n, 1, 10000}] ta=Delete[ta, 1]
Prime[DivisorSigma[1, #]]&/@Select[Range[5000], Prime[DivisorSigma[ 1, #]] == Prime[ EulerPhi[#]]+6&] (* Harvey P. Dale, Sep 22 2016 *)

Prog

(PARI) p=2; q=3; forprime(r=5, 1e6, if(r-p==6 && isprime(primepi(q)), print1(r", ")); p=q; q=r) \\ Charles R Greathouse IV, May 15 2013

Crossrefs

Cf. A067161, A048848, A000020, A000203.

Sequence in context: A106903 A098413 A174152 * A156940 A220162 A248483

Adjacent sequences: A103173 A103174 A103175 * A103177 A103178 A103179

Keyword

nonn

Author

Labos Elemer, Mar 02 2005

Extensions

a(1) corrected by Charles R Greathouse IV, May 15 2013

Status

approved