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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.)

%I A243003 #29 Sep 05 2019 00:03:12
%S A243003 2,4,4,5,11,13,16,17,16,29,57,78
%N 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.)
%C A243003 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 A243003 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 A243003 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 A243003 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 A243003 One must have p/2 < k < p and (p-k) | (k-1).
%e A243003 For given p=A000043(n), the following k's yield a prime R:
%e A243003 p : k's
%e A243003 2 : -
%e A243003 3 : 2
%e A243003 5 : 4
%e A243003 7 : 4, 5
%e A243003 13 : 11
%e A243003 17 : 13
%e A243003 19 : 16, 17
%e A243003 31 : 16, 29
%e A243003 61 : 57
%e A243003 89 : 78
%e A243003 107 through 86243 : none.
%e A243003 107 through 3021377: none. - _Robert Price_, Sep 04 2019
%e A243003 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 A243003 A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
%t A243003    1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
%t A243003    21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
%t A243003    1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
%t A243003    24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
%t A243003    43112609};
%t A243003 lst = {};
%t A243003 For[i = 1, i <= 10, i++,
%t A243003   p = A000043[[i]];
%t A243003   For[k = 1, k < p, k++,
%t A243003    r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
%t A243003    If[! IntegerQ[r], Continue[]];
%t A243003    If[PrimeQ[r], AppendTo[lst, k]]]];
%t A243003 lst (* _Robert Price_, Sep 04 2019 *)
%o A243003 (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 A243003 Cf. A258882 (PWN of the form 2^k*p*q).
%Y A243003 Cf. A242993 (least k), A242998 (number of solution for given p in A000043), A242999 (values of p), A242025 list of all primes R.
%K A243003 nonn,hard,more
%O A243003 1,1
%A A243003 _M. F. Hasler_, Aug 17 2014