A301806 -id:A301806 - cp's OEIS frontend

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

Paolo Galliani, Jan 23 2018

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

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?.

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

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

[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

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

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

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

A378288 Array read by antidiagonals: row k consists of the positive integers j for which the concatenation of 2^k - 1 and 2^j - 1 is prime.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 5, 5, 2, 1, 6, 9, 7, 3, 1, 9, 15, 10, 5, 2, 1, 13, 27, 11, 9, 3, 7, 3, 18, 45, 13, 15, 6, 17, 6, 1, 19, 59, 14, 45, 9, 41, 13, 3, 2, 26, 211, 25, 61, 11, 101, 125, 7, 5, 13, 43, 303, 62, 65, 13, 157, 150, 9, 11, 27, 3, 46, 425, 70, 227, 23, 367, 195, 11, 14, 43, 14, 11

Offset: 1

Robert Israel, Nov 26 2024

No terms are divisible by 4.
All terms in row k are coprime to k.
Conjecture: all rows have infinitely many terms, and all positive integers not divisible by 4 appear in infinitely many rows.
k - 1 is in row k iff k is in A301806.

			The array starts
   1    2    3    5    6    9   13   18 ...
   1    3    5    9   15   27   45   59 ...
   1    2    7   10   11   13   14   25 ...
   1    3    5    9   15   45   61   65 ...
   1    2    3    6    9   11   13   23 ...
   1    7   17   41  101  157  367  571 ...
   3    6   13  125  150  195  634 1282 ...
   1    3    7    9   11   23   27   39 ...
a(3,4) = 10 is a term in row 3 because the concatenation of 2^3 - 1 = 7 and 2^10 - 1 = 1023 is 71023, which is prime.

Cf. A000225, A301806.

tcat:= (a,b) -> 10^(1+ilog10(b))*a+b:
N:= 8: # for the top left N x N array
M:= Matrix(N, N):
for i from 1 to N do
   count:= 0:
   x:= 2^i-1;
   for j from 1 by `if`(i::even,2,1) while count + i < N do
     if j mod 4 = 0 or igcd(i,j) > 1 then next fi;
     if isprime(tcat(x,2^j-1)) then count:= count+1; M[i,count]:= j fi;
   od;
od:
M;
seq(seq(M[k,1+d-k], k=1..d), d=1..N-1);

cp's OEIS Frontend

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

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