A298613 Primes formed by the concatenation of 2^k-1 and 2^(k-1)-1.

31, 73, 157, 12763, 255127, 40952047, 524287262143, 41943032097151, 6871947673534359738367, 7036874417766335184372088831, 22517998136852471125899906842623, 14757395258967641292773786976294838206463, 604462909807314587353087302231454903657293676543

Offset: 1

Author

Paolo Galliani, Jan 23 2018

Keywords

• nonn
• base

Comments

Conjectures:

(1) The factorization of a(n) + 1 never contains an odd prime squared.

(2) a(n) + 1 is not divisible by 7.

(3) There are infinitely many primes of this form.

(4) The fifth term, 255127 is the only term of the sequence which can be written as the sum of a square and a repunit. In fact, 255127=504^2+1111.

(5) The fifth term, 255127 is the only term of the sequence which is congruent to 1 mod 101.

(6) a(9) is the largest term of the sequence for which k is a power.

Note: a(n) can never be a Sophie Germain prime. - Max Alekseyev, Mar 30 2018

Note that from a(20) to a(28), the primes are congruent to 3 (mod 7), whereas a(30), a(31), a(32), a(33) and a(34) are all congruent to 5 (mod 7). - Paolo Galliani, Jun 17 2018 and Jun 25 2018

The first conjecture has been disproved. In fact, a(36)+1 is divisible by 23^2. - Paolo Galliani, Aug 27 2018

The first four terms of the sequence: 31, 73, 157, 12763 are emirps. - Paolo Galliani, Nov 05 2018

The first four terms of the sequence reversed: 13, 37, 751, 36721 are Chen primes. - Paolo Galliani, Nov 09 2018

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..18
• MathOverflow, Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?.

Crossrefs

Cf. A000040, A000225, A005384, A301806, A006567.

Programs

GAP

m:=300;;
g1:=List(List([1..m],k->2^k-1),ListOfDigits);;
g2:=List(List([1..m],k->2^(k-1)-1),ListOfDigits);;
g3:=List([1..m],i->Concatenation(g1[i],g2[i]));;
a:=Filtered(List([1..Length(g3)],s->Sum([0..Length(g3[s])-1],t->g3[s][Length(g3[s])-t]*10^t)),IsPrime); # Muniru A Asiru, Mar 29 2018

Magma

[t: n in [1..100] | IsPrime(t) where t is Seqint(Intseq(2^(n-1)-1) cat Intseq(2^n-1))]; // Bruno Berselli, Feb 02 2018

Mathematica

Select[Map[#1 10^IntegerLength@ #2 + #2 & @@ Reverse@ # &, Partition[Array[2^# - 1 &, 90], 2, 1]], PrimeQ] (* Michael De Vlieger, Jan 23 2018 *)

PARI

lista(nn) = for (n=1, nn, if (isprime(p=fromdigits(concat(digits(2^n-1), digits(2^(n-1)-1)))), print1(p, ", "))); \\ Michel Marcus, Jan 29 2018

Formula

a(n) = concatenation of 2^k-1 and 2^(k-1)-1, where k = A301806(n).