A242999 Mersenne prime exponents p in A000043 such that R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime for some k < p, listed with multiplicity (number of k's), see A243003 for the k-values.

3, 5, 7, 7, 13, 17, 19, 19, 31, 31, 61, 89

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. In the special case of Mersenne primes Q = 2^p-1, p = A000043(n), considered here, one has R = 2^k-1+(2^k-2)/(2^(p-k)-1).

This sequence lists the p-values. See sequence A243003 for the k-values and A242998(n) for the number of possible k-values for a given p = A000043(n), i.e., the number of times this p appears here.

The next term, a(13), is larger than 80000 (if it exists).

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

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 first component of these pairs, sequence A243003 lists the second component.

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, p]]]];
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(p", "))) \\ M. F. Hasler, Jul 19 2016

Crossrefs

Sequence in context: A060265 A172365 A297709 * A098566 A006540 A099726

Adjacent sequences: A242996 A242997 A242998 * A243000 A243001 A243002

Keyword

nonn,hard,more

Author

M. F. Hasler, Aug 17 2014

Status

approved