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

%I A298613 #240 Mar 11 2024 01:54:08
%S A298613 31,73,157,12763,255127,40952047,524287262143,41943032097151,
%T A298613 6871947673534359738367,7036874417766335184372088831,
%U A298613 22517998136852471125899906842623,14757395258967641292773786976294838206463,604462909807314587353087302231454903657293676543
%N A298613 Primes formed by the concatenation of 2^k-1 and 2^(k-1)-1.
%C A298613 Conjectures:
%C A298613 (1) The factorization of a(n) + 1 never contains an odd prime squared.
%C A298613 (2) a(n) + 1 is not divisible by 7.
%C A298613 (3) There are infinitely many primes of this form.
%C A298613 (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.
%C A298613 (5) The fifth term, 255127 is the only term of the sequence which is congruent to 1 mod 101.
%C A298613 (6) a(9) is the largest term of the sequence for which k is a power.
%C A298613 Note: a(n) can never be a Sophie Germain prime. - _Max Alekseyev_, Mar 30 2018
%C A298613 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
%C A298613 The first conjecture has been disproved. In fact, a(36)+1 is divisible by 23^2. - _Paolo Galliani_, Aug 27 2018
%C A298613 The first four terms of the sequence: 31, 73, 157, 12763 are emirps. - _Paolo Galliani_, Nov 05 2018
%C A298613 The first four terms of the sequence reversed: 13, 37, 751, 36721 are Chen primes. - _Paolo Galliani_, Nov 09 2018
%H A298613 Muniru A Asiru, <a href="/A298613/b298613.txt">Table of n, a(n) for n = 1..18</a>
%H A298613 MathOverflow, <a href="https://mathoverflow.net/questions/294527/are-concatenations-of-two-consecutive-mersenne-numbers-which-are-congruent-to-6">Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?</a>.
%F A298613 a(n) = concatenation of 2^k-1 and 2^(k-1)-1, where k = A301806(n).
%t A298613 Select[Map[#1 10^IntegerLength@ #2 + #2 & @@ Reverse@ # &, Partition[Array[2^# - 1 &, 90], 2, 1]], PrimeQ] (* _Michael De Vlieger_, Jan 23 2018 *)
%o A298613 (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
%o A298613 (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
%o A298613 (GAP) m:=300;;
%o A298613 g1:=List(List([1..m],k->2^k-1),ListOfDigits);;
%o A298613 g2:=List(List([1..m],k->2^(k-1)-1),ListOfDigits);;
%o A298613 g3:=List([1..m],i->Concatenation(g1[i],g2[i]));;
%o A298613 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
%Y A298613 Cf. A000040, A000225, A005384, A301806, A006567.
%K A298613 nonn,base
%O A298613 1,1
%A A298613 _Paolo Galliani_, Jan 23 2018