A243003 Pairs (p,k) such that p is in A000043 and R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime: this sequence lists the k-values, see A242999 for the p-values. (Ordered by p, then k.)

2, 4, 4, 5, 11, 13, 16, 17, 16, 29, 57, 78

Offset

1,1

Comments

Related to the search for large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf A002975) when Q > 2^k and R = (2^k*Q-Q-1)/(Q+1-2^k) both are prime. Here we consider the special case where Q = 2^p-1 is a Mersenne prime, p = A000043(n). For such Q one has R = 2^k-1+(2^k-2)/(2^(p-k)-1). [First R formula corrected by Jens Kruse Andersen, Aug 18 2014]

This sequence lists the k-values, see sequence A242999 for the p-values and A242998(n) for the number of possible k-values for given p = A000043(n) resp. Q = A000668(n).

This sequence can also be considered as a table whose n-th row holds the possible k-values for the n-th Mersenne prime Q = A000668(n); sequence A242998 gives the row lengths of the table (which are mostly 0).

Links

Table of n, a(n) for n=1..12.

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

Formula

One must have p/2 < k < p and (p-k) | (k-1).

Example

For given p=A000043(n), the following k's yield a prime R:
p : k's
2 : -
3 : 2
5 : 4
7 : 4, 5
13 : 11
17 : 13
19 : 16, 17
31 : 16, 29
61 : 57
89 : 78
107 through 86243 : none.
107 through 3021377: none. - Robert Price, Sep 04 2019
Thus the pairs (p,k) are (3,2), (5,4), (7,4), (7,5), (13,11), ... and the present sequence lists the second component of these pairs, the first components are listed in A242999.

Mathematica

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 10, i++,
  p = A000043[[i]];
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], AppendTo[lst, k]]]];
lst (* Robert Price, Sep 04 2019 *)

Prog

(PARI) forprime(p=1, , ispseudoprime(2^p-1)||next; for(k=p\2+1, p-1, (k-1)%(p-k)==0 && isprime(2^k-1+(2^k-2)/(2^(p-k)-1))&&print1(k", "))) \\ M. F. Hasler, Jul 19 2016

Crossrefs

Cf. A258882 (PWN of the form 2^k*p*q).

Cf. A242993 (least k), A242998 (number of solution for given p in A000043), A242999 (values of p), A242025 list of all primes R.

Sequence in context: A366541 A327629 A121528 * A223227 A160904 A306340

Adjacent sequences: A243000 A243001 A243002 * A243004 A243005 A243006

Keyword

nonn,hard,more

Author

M. F. Hasler, Aug 17 2014

Status

approved