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

Author

Labos Elemer, Mar 02 2005

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A067161, A048848, A000020, A000203.

Programs

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 *)

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

Extensions

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