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A180642 Numbers k such that phi(k)/4 is a prime, where phi is the Euler totient function.

%I A180642 #20 Mar 08 2020 01:39:49
%S A180642 13,15,16,20,21,24,25,26,28,29,30,33,36,42,44,50,53,58,66,69,92,106,
%T A180642 138,141,149,173,177,188,236,249,269,282,293,298,317,321,332,346,354,
%U A180642 389,428,498,501,509,537,538,557,586,634,642,653,668,681,716,773,778,789
%N A180642 Numbers k such that phi(k)/4 is a prime, where phi is the Euler totient function.
%C A180642 Apparently the sequence is infinite, but I have no proof. There are many n-ples of consecutives: (15,16)-(20,21)-(24,25,26)-(537,538)-(1436,1437)-...-(30236-30237)
%C A180642 This sequence is infinite if and only if there are infinitely many primes of the form 2p+1 or 4p+1 with prime p. - _Charles R Greathouse IV_, Feb 04 2013
%H A180642 Amiram Eldar, <a href="/A180642/b180642.txt">Table of n, a(n) for n = 1..10000</a>
%F A180642 a(n) >> n log^2 n. - _Charles R Greathouse IV_, Feb 04 2013
%e A180642 a(5) = 21 since pi(21)/4 = 12/4 = 3 is prime.
%t A180642 Select[Range[800],PrimeQ[EulerPhi[#]/4]&] (* _Harvey P. Dale_, Feb 11 2015 *)
%o A180642 (PARI) is(n)=n=eulerphi(n);n%4==0 && isprime(n/4) \\ _Charles R Greathouse IV_, Feb 04 2013
%o A180642 (PARI) is(n)=if(n<51,n=eulerphi(n);n%4==0 && isprime(n/4),my(v=[3,4,6]);for(i=1,#v,if(n%(2*v[i])==v[i]&&gcd(n/v[i],v[i])==1&&isprime(n/v[i])&&isprime(eulerphi(n)/4),return(1)));if(n%4==2,n/=2);n%4==1&&isprime(n)&&isprime(n\4)) \\ _Charles R Greathouse IV_, Feb 04 2013
%Y A180642 Cf. A000010, A065966 (phi(k)/2 is prime), A090866 (subsequence of primes).
%K A180642 nonn
%O A180642 1,1
%A A180642 _Carmine Suriano_, Sep 14 2010