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

1,1

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?.

Formula

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

Mathematica

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

Prog

(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
(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
(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

Crossrefs

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

Sequence in context: A083988 A070954 A141892 * A155933 A163428 A130468

Adjacent sequences: A298610 A298611 A298612 * A298614 A298615 A298616

Keyword

nonn,base

Author

Paolo Galliani, Jan 23 2018

Status

approved