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A242998
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Number of integers k such that R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) is a prime number, when Q = A000668(n) is the n-th Mersenne prime.
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7
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0, 1, 1, 2, 1, 1, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,4
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COMMENTS
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Related to the search of 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 count such primes for 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).
See A242025 for the resulting primes R, which however are there not listed in order of the p's.
This sequence gives the row lengths for the table A243003 whose rows hold the k-values leading to prime R, for a given Mersenne prime.
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LINKS
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S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003
E. Weisstein, Weird numbers, on MathWorld - a Wolfram web ressource.
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EXAMPLE
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For given p=A000043(n), the following k's yield a prime R:
p : k's (and resulting primes R, Q=2^p-1 and/or weird W=2^(k-1)*Q*R)
2 : -
3 : 2 (R=5, Q=7, W=70)
5 : 4 (R=29, Q=31, W=7192)
7 : 4 (R=17, Q=127, W=17272), 5 (R=41, Q=127, W=83312)
13 : 11 (R=2729, Q=8191, W=22889716736)
17 : 13 (R=8737, Q=131071, W=4690605371392)
19 : 16 (R=74897, W=1286718208049152), 17 (R=174761, W=6004730783793152)
31 : 16 (R=65537, W=2^15*(2^31-1)*R), 29 (R=715827881, W=2^28*(2^31-1)*R)
61 : 57 (R=153722867280912929, W=2^56*(2^61-1)*R)
89 : 78 (R=302379100949042568368129, W=2^77*(2^89-1)*R)
107 through 86243 : none.
The present sequence lists the number of k's in each line.
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MATHEMATICA
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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 <= 28, i++,
kc = 0;
For[k = 1, k < p, k++,
r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
If[! IntegerQ[r], Continue[]];
If[PrimeQ[r], kc++]];
AppendTo[lst, kc]];
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PROG
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(PARI) A242998(n, p=A000043[n])={sum(k=p\2+1, p-1, Mod(2, 2^(p-k)-1)^k==2 && ispseudoprime(2^k-1+(2^k-2)/(2^(p-k)-1)))}
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CROSSREFS
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See also A320875 for more general solutions to R = (MQ-1)/(Q-M) = prime.
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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