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

M. F. Hasler, Table of n, a(n) for n = 1..12

CWU press release, CWU Math Students Break World Record for Largest Weird Number, Dec. 4, 2013.

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

KEYWORD

nonn,hard,more

AUTHOR

M. F. Hasler, Aug 17 2014

STATUS

approved