A247024 Primes of the form x*(2^x - x) - 1.

3, 47, 347, 10139, 396527668833598369303619003, 1624796301562061610805093487, 2637188343637273091841153207596203623407

Offset: 1

Author

R. J. Cano, Sep 09 2014

Keywords

• nonn

Comments

Subsequence of A245014. a(n) represents the identity between (p + 4k^2 + 1) and (2n*4^k) for the least prime p defining A245014(k). Those first k where it occurs are: 1,2,3,5,41,42,62,183,357,407.

Consider the table of Stirling numbers of the second kind (A008277). The second column contains the numbers A000225, or 2^m - 1, and the first subdiagonal contains the triangular numbers. If a number appears in both sequences, we have the equation f(x) = 2^x - x^2 + x - 2 = 0 which has integer roots x = 1, 2, 3. Set g(x) = (x - 1)*(2^x - 1). Then it is found that the sum f(x) + g(x) for some even x defines this sequence and satisfies in common with A245014: Both sequences have three consecutive terms (those first) such that when they are represented in decimal the third term is the concatenation of the two terms preceding it.

Prime or PRP for x = 2, 4, 6, 10, 82, 84, 124, 366, 714, 814, 1584, 8938, 17812, 27054, 35380, 71358. - Jens Kruse Andersen, Sep 10 2014

The complete solution to the remark on Stirling2 numbers in a comment above is given in A076046. See also my Oct 08 2014 remark in the history. - Wolfdieter Lang, Oct 16 2014

Links

• R. J. Cano, Table of n, a(n) for n = 1..11

Crossrefs

Cf. A000225, A008277, A245014.

Programs

Magma

[a: x in [0..450] | IsPrime(a) where a is x*(2^x - x)-1]; // Vincenzo Librandi, Oct 16 2014

Maple

A247024:=n->`if`(isprime(n*(2^n-n)-1),n*(2^n-n)-1,NULL): seq(A247024(n), n=1..10^2); # Wesley Ivan Hurt, Sep 13 2014

Mathematica

Select[Table[n(2^n - n) - 1, {n, 200}], PrimeQ] (* Alonso del Arte, Sep 09 2014 *)

PARI

listToVec(l:list,Comp,Omega=#l)=vector(Omega,i,l[i][Comp]);
fplusg(x)=x*(2^x-x)-1;
find_next_N_terms(N,resumeFrom:list=List(),verbose=0)= {
  my(s:list,q,j,F,N0);
  s=resumeFrom;
  q=#s;
  N0=q;
  if(q,j=s[q][1]);
  while(q
				

PFGW

ABC2 $a*(2^$a-$a)-1
a: from 2 to 100000 step 2
Charles R Greathouse IV, Sep 18 2014

Extensions

Edited: 'penultimate column' replaced by 'first subdiagonal'. - Wolfdieter Lang, Oct 16 2014