A243003 - OEIS

%I #29 Sep 05 2019 00:03:12

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

%N 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.)

%C 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]

%C 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).

%C 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).

%H S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). <a href="http://zbmath.org/?format=complete&amp;q=an:0365.10003">Zbl 0365.10003</a>

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

%e For given p=A000043(n), the following k's yield a prime R:

%e p : k's

%e 2 : -

%e 3 : 2

%e 5 : 4

%e 7 : 4, 5

%e 13 : 11

%e 17 : 13

%e 19 : 16, 17

%e 31 : 16, 29

%e 61 : 57

%e 89 : 78

%e 107 through 86243 : none.

%e 107 through 3021377: none. - _Robert Price_, Sep 04 2019

%e 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.

%t A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,

%t    1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,

%t    21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,

%t    1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,

%t    24036583, 25964951, 30402457, 32582657, 37156667, 42643801,

%t    43112609};

%t lst = {};

%t For[i = 1, i <= 10, i++,

%t   p = A000043[[i]];

%t   For[k = 1, k < p, k++,

%t    r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);

%t    If[! IntegerQ[r], Continue[]];

%t    If[PrimeQ[r], AppendTo[lst, k]]]];

%t lst (* _Robert Price_, Sep 04 2019 *)

%o (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

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

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

%K nonn,hard,more

%O 1,1

%A _M. F. Hasler_, Aug 17 2014