cp's OEIS Frontend

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.

%I A242999 #28 Sep 05 2019 00:03:05
%S A242999 3,5,7,7,13,17,19,19,31,31,61,89
%N 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.
%C A242999 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).
%C A242999 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.
%C A242999 The next term, a(13), is larger than 80000 (if it exists).
%H A242999 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>
%e A242999 For given p = A000043(n), the following k's yield a prime R:
%e A242999 p : k's
%e A242999 2 : -
%e A242999 3 : 2
%e A242999 5 : 4
%e A242999 7 : 4, 5
%e A242999 13 : 11
%e A242999 17 : 13
%e A242999 19 : 16, 17
%e A242999 31 : 16, 29
%e A242999 61 : 57
%e A242999 89 : 78
%e A242999 107 through 86243 : none.
%e A242999 107 through 3021377: none. - _Robert Price_, Sep 04 2019
%e A242999 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.
%t A242999 A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
%t A242999    1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
%t A242999    21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
%t A242999    1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
%t A242999    24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
%t A242999    43112609};
%t A242999 lst = {};
%t A242999 For[i = 1, i <= 10, i++,
%t A242999   p = A000043[[i]];
%t A242999   For[k = 1, k < p, k++,
%t A242999    r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
%t A242999    If[! IntegerQ[r], Continue[]];
%t A242999    If[PrimeQ[r], AppendTo[lst, p]]]];
%t A242999 lst (* _Robert Price_, Sep 04 2019 *)
%o A242999 (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
%K A242999 nonn,hard,more
%O A242999 1,1
%A A242999 _M. F. Hasler_, Aug 17 2014