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%I #44 May 06 2020 08:35:04
%S 7,13,17,19,43,47,59,61,71,79,101,107,109,149,151,163,167,197,223,257,
%T 263,271,311,317,347,349,353,383,389,401,421,439,449,461,479,503,521,
%U 523,557,569,599,601,613,631,673,677,691,701,811,821,827,839,853,863,881,919
%N Primes p such that d(p*(2p+1)) = 8 where d(n) is the number of divisors of n (A000005).
%C Primes p such that 2p+1 is in A030513. - _Robert Israel_, Aug 17 2016
%C From _Anthony Hernandez_, Aug 29 2016: (Start)
%C Conjecture: this sequence is infinite.
%C It appears that the prime numbers in this sequence which have 7 for as final digit form the sequence A104164.
%C Conjecture: this sequence contains infinitely many twin primes. The first few twin primes in this sequence are 17,19,59,61,107,109,521,523,599,601,... (End)
%C From _Bernard Schott_, Apr 28 2020: (Start)
%C This sequence equals the union of {13} and A234095; proof by double inclusion:
%C -> 1st inclusion: {13} Union A234095 is included in A276045.
%C 1) if p = 13, then 13*27 = 351 = 3^3 * 13, hence d(351) = 8 and 13 belongs to A276045.
%C 2) if p is in A234095, then p*(2*p+1) = p*r*s (p,r,s primes) and d(p*r*s) = 8, hence p is in 276045.
%C -> 2nd inclusion: A276045 is included in {13} Union A234095.
%C If p is in A276095, then m=p*(2*p+1) has 8 divisors and there are only three possibilities: m = u*v*w, or m = u^3*v or m = u^7 with u, v, w are distinct primes.
%C 1st case: if p*(2*p+1) = u*v*w then u=p, and 2p+1=v*w is semiprime; hence, p is in A234095 Union {13}.
%C 2nd case: if p*(2p+1) = u^3*v then p=v and 2*p+1=u^3 ==> 2*p = u^3-1 = (u-1)*(u^2+u+1) with 2 and p are primes; then (u-1=2, u^2+u+1=p) so u=3, and p=3^2+3+1=13; hence p = 13 belongs to {13} Union A234095.
%C 3rd case: p*(2p+1) = u^7 is impossible.
%C Conclusion: this sequence = {13} Union A234095. (End)
%H Amiram Eldar, <a href="/A276045/b276045.txt">Table of n, a(n) for n = 1..10000</a>
%e d(7*(2*7+1))=d(105)=8 so 7 is a term.
%p select(n -> isprime(n) and numtheory:-tau(n*(2*n+1))=8,
%p [seq(i, i=3..1000, 2)]); # _Robert Israel_, Aug 17 2016
%t Select[Prime@ Range@ 160, DivisorSigma[0, # (2 # + 1)] == 8 &] (* _Michael De Vlieger_, Aug 28 2016 *)
%o (PARI) lista(nn) = forprime(p=2, nn, if (numdiv(p*(2*p+1))==8, print1(p, ", "))); \\ _Michel Marcus_, Aug 17 2016
%Y Cf. A000005, A030513, A030626.
%Y Equals {13} Union A234095.
%K nonn,easy
%O 1,1
%A _Anthony Hernandez_, Aug 17 2016
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