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

Author

M. F. Hasler, Aug 17 2014

Keywords

• nonn
• hard
• more

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).

Examples

			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.
		

Links

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

Programs

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 *)

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