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%I #15 May 23 2021 02:43:11
%S 13,19,43,113,463,619,863,1789,2273,2383,4519,4789,4937,5443,5507,
%T 5653,8237,10459,13007,13697,16063,16453,17389,18313,18919,20903,
%U 21193,21319,21383,23567,24109,25309,26267,27947,28283,29573,30559,31183,31517
%N Let p = prime(sigma(n)) and q = prime(phi(n)), then p is in the sequence if p-q = 6.
%C Conjecture: In all cases sigma(n)-phi(n)=2, i.e., n is prime.
%C 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
%H Charles R Greathouse IV, <a href="/A103176/b103176.txt">Table of n, a(n) for n = 1..10000</a>
%e n=3719, sigma(n)=3720, phi(n)=3718, a(n)=p(sigma(n))=34847.
%t 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]
%t Prime[DivisorSigma[1,#]]&/@Select[Range[5000],Prime[DivisorSigma[ 1,#]] == Prime[ EulerPhi[#]]+6&] (* _Harvey P. Dale_, Sep 22 2016 *)
%o (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
%Y Cf. A067161, A048848, A000020, A000203.
%K nonn
%O 1,1
%A _Labos Elemer_, Mar 02 2005
%E a(1) corrected by _Charles R Greathouse IV_, May 15 2013
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