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A242025 Primes of the form R = 2^k-1+(2^k-2)/(2^(p-k)-1), where p are Mersenne prime exponents listed in A000043.

%I A242025 #60 Sep 05 2019 00:02:56
%S A242025 5,17,29,41,2729,8737,65537,74897,174761,715827881,153722867280912929,
%T A242025 302379100949042568368129
%N A242025 Primes of the form R = 2^k-1+(2^k-2)/(2^(p-k)-1), where p are Mersenne prime exponents listed in A000043.
%C A242025 Related to the search for large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number when Q > 2^k and R = (2^k*Q-Q-1)/(Q+1-2^k) both are prime, cf. subset A258882 of A002975. Here we consider such primes for the special case of Mersenne primes Q = 2^p-1, p in A000043. For such Q one has R = 2^k-1+(2^k-2)/(2^(p-k)-1), which must be an integer and prime number.
%C A242025 See A242998 for the number of exponents k leading to primes R, for given Q = A000668(n) = 2^p-1, p = A000043(n). But there is no one-to-one correspondence since the primes R are here listed according to their size (cf. example). The pairs (k,p) are given in A242999 and A243003.
%C A242025 Kravitz used his formula in 1976 to find the 53-digit PWN corresponding to a(11), cf. examples. In 2013, students of CWU used the same idea to find the next term in the series, corresponding to a(12), see examples. They found still larger PWN of the same form with other primes Q, see A320875. This renewed the interest in weird numbers and motivated several recent papers, cf. A002975. - _M. F. Hasler_, Nov 10 2018
%H A242025 M. F. Hasler, <a href="/A242025/b242025.txt">Table of n, a(n) for n = 1..12</a>
%H A242025 CWU press release, <a href="http://www.cwu.edu/cwu-math-students-break-world-record-largest-weird-number-0">CWU Math Students Break World Record for Largest Weird Number</a>, Dec. 4, 2013.
%H A242025 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 A242025 For given p = A000043(n), the following k yield a prime R and an associated (primitive) weird number W = 2^(k-1)*(2^p-1)*R in A258882 c A002975 c A006037:
%e A242025 For p = 2, no k yields a prime R = 2^k-1+(2^k-2)/(2^(p-k)-1).
%e A242025 For p = 3, k = 2 yields R = 5 and the (smallest) weird number W = 70 = A006037(1).
%e A242025 For p = 5, k = 4 yields R = 29 = a(3) and W = 7192 = A258882(3).
%e A242025 For p = 7, k = 4 yields R = 17 = a(2) and W = 17272 = A258882(7),
%e A242025        and k = 5 yields R = 41 = a(4) and W = 83312 = A258882(9).
%e A242025 For p = 13, k = 11 yields R = 2729 = a(5) and W = 22889716736 = A258882(288)
%e A242025 For p = 17, k = 13 yields R = 8737 = a(6) and W = 4690605371392 = A258882(1203).
%e A242025 For p = 19, k = 16 yields R = 74897 = a(8), W = 1286718208049152 = A258882(7154),
%e A242025         and k = 17 yields R = 174761 = a(9), W = 6004730783793152 = A258882(11466).
%e A242025 For p = 31, k = 16 yields R = 65537 = a(7) (smaller than both R's for p = 19),
%e A242025         and k = 29 yields R = 715827881 = a(10).
%e A242025 For p = 61, only k = 57 yields a prime R = 153722867280912929 = a(11).
%e A242025 For p = 89, only k = 78 yields a prime R = 302379100949042568368129 = a(12).
%e A242025 For p = 107 through p = 86243, no k yields a prime R.
%e A242025 For p = 107 through p = 3021377, no k yields a prime R. - _Robert Price_, Sep 04 2019
%t A242025 A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
%t A242025    1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
%t A242025    21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
%t A242025    1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
%t A242025    24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
%t A242025    43112609};
%t A242025 lst = {};
%t A242025 For[i = 1, i <= Length[A000043], i++,
%t A242025   p = A000043[[i]];
%t A242025   For[k = 1, k < p, k++,
%t A242025    r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
%t A242025    If[! IntegerQ[r], Continue[]];
%t A242025    If[PrimeQ[r], AppendTo[lst, r]]]];
%t A242025 Union[lst] (* _Robert Price_, Sep 04 2019 *)
%Y A242025 Cf. A258882 (PWN of the form 2^k*p*q).
%Y A242025 Cf. A242993, A242998, A242999 and A243003: related to PWN with a prime factor R as defined here, with Q = A000668(n) = 2^A000043(n)-1.
%Y A242025 Cf. A320875 for a more general pattern leading to more and larger PWN.
%K A242025 nonn,hard,more
%O A242025 1,1
%A A242025 _M. F. Hasler_, Aug 17 2014