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A298613 Primes formed by the concatenation of 2^k-1 and 2^(k-1)-1. 2
31, 73, 157, 12763, 255127, 40952047, 524287262143, 41943032097151, 6871947673534359738367, 7036874417766335184372088831, 22517998136852471125899906842623, 14757395258967641292773786976294838206463, 604462909807314587353087302231454903657293676543 (list; graph; refs; listen; history; text; internal format)
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
FORMULA
a(n) = concatenation of 2^k-1 and 2^(k-1)-1, where k = A301806(n).
MAPLE
P:=proc(n) local a;
a:=2^(n-1)-1+(2^n-1)*10^(ilog10(2^(n-1)-1)+1); if isprime(a) then a;
fi; end: seq(P(i), i=2..10^2); # Paolo P. Lava, Jan 23 2018
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
Sequence in context: A083988 A070954 A141892 * A155933 A163428 A130468
KEYWORD
nonn,base
AUTHOR
Paolo Galliani, Jan 23 2018
STATUS
approved

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Last modified February 21 22:50 EST 2024. Contains 370239 sequences. (Running on oeis4.)