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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.
6
5, 17, 29, 41, 2729, 8737, 65537, 74897, 174761, 715827881, 153722867280912929, 302379100949042568368129
OFFSET
1,1
COMMENTS
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.
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.
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
LINKS
S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003
EXAMPLE
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:
For p = 2, no k yields a prime R = 2^k-1+(2^k-2)/(2^(p-k)-1).
For p = 3, k = 2 yields R = 5 and the (smallest) weird number W = 70 = A006037(1).
For p = 5, k = 4 yields R = 29 = a(3) and W = 7192 = A258882(3).
For p = 7, k = 4 yields R = 17 = a(2) and W = 17272 = A258882(7),
and k = 5 yields R = 41 = a(4) and W = 83312 = A258882(9).
For p = 13, k = 11 yields R = 2729 = a(5) and W = 22889716736 = A258882(288)
For p = 17, k = 13 yields R = 8737 = a(6) and W = 4690605371392 = A258882(1203).
For p = 19, k = 16 yields R = 74897 = a(8), W = 1286718208049152 = A258882(7154),
and k = 17 yields R = 174761 = a(9), W = 6004730783793152 = A258882(11466).
For p = 31, k = 16 yields R = 65537 = a(7) (smaller than both R's for p = 19),
and k = 29 yields R = 715827881 = a(10).
For p = 61, only k = 57 yields a prime R = 153722867280912929 = a(11).
For p = 89, only k = 78 yields a prime R = 302379100949042568368129 = a(12).
For p = 107 through p = 86243, no k yields a prime R.
For p = 107 through p = 3021377, no k yields a prime R. - Robert Price, Sep 04 2019
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 <= Length[A000043], 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, r]]]];
Union[lst] (* Robert Price, Sep 04 2019 *)
CROSSREFS
Cf. A258882 (PWN of the form 2^k*p*q).
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.
Cf. A320875 for a more general pattern leading to more and larger PWN.
Sequence in context: A074965 A145475 A071695 * A027703 A075695 A146278
KEYWORD
nonn,hard,more
AUTHOR
M. F. Hasler, Aug 17 2014
STATUS
approved