A276045 Primes p such that d(p*(2p+1)) = 8 where d(n) is the number of divisors of n (A000005).

7, 13, 17, 19, 43, 47, 59, 61, 71, 79, 101, 107, 109, 149, 151, 163, 167, 197, 223, 257, 263, 271, 311, 317, 347, 349, 353, 383, 389, 401, 421, 439, 449, 461, 479, 503, 521, 523, 557, 569, 599, 601, 613, 631, 673, 677, 691, 701, 811, 821, 827, 839, 853, 863, 881, 919

Offset: 1

Author

Anthony Hernandez, Aug 17 2016

Keywords

• nonn
• easy

Comments

Primes p such that 2p+1 is in A030513. - Robert Israel, Aug 17 2016

From Anthony Hernandez, Aug 29 2016: (Start)

Conjecture: this sequence is infinite.

It appears that the prime numbers in this sequence which have 7 for as final digit form the sequence A104164.

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)

From Bernard Schott, Apr 28 2020: (Start)

This sequence equals the union of {13} and A234095; proof by double inclusion:

-> 1st inclusion: {13} Union A234095 is included in A276045.

1) if p = 13, then 13*27 = 351 = 3^3 * 13, hence d(351) = 8 and 13 belongs to A276045.

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.

-> 2nd inclusion: A276045 is included in {13} Union A234095.

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.

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}.

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.

3rd case: p*(2p+1) = u^7 is impossible.

Conclusion: this sequence = {13} Union A234095. (End)

Examples

			d(7*(2*7+1))=d(105)=8 so 7 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A030513, A030626.

Equals {13} Union A234095.

Programs

Maple

select(n -> isprime(n) and numtheory:-tau(n*(2*n+1))=8,
[seq(i, i=3..1000, 2)]); # Robert Israel, Aug 17 2016

Mathematica

Select[Prime@ Range@ 160, DivisorSigma[0, # (2 # + 1)] == 8 &] (* Michael De Vlieger, Aug 28 2016 *)

PARI

lista(nn) = forprime(p=2, nn, if (numdiv(p*(2*p+1))==8, print1(p, ", "))); \\ Michel Marcus, Aug 17 2016