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A318291
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a(n) is the minimum k > 0 such that n*2^k - 3 is prime, or 0 if no such k exists.
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3
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3, 2, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 2, 1, 0, 1, 2, 0, 1, 3, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 2, 7, 0, 3, 1, 0, 1, 2, 0, 1, 1, 0, 5, 2, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 1, 4, 0
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OFFSET
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1,1
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COMMENTS
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Question: Other than multiples of 3, do there exist any numbers n > 3 such that a(n) = 0?
From Robert Israel, Aug 24 2018: (Start)
The answer is yes. The situation is similar to that of Riesel or Sierpinski numbers.
Every integer k is in at least one of the following residue classes:
2 (mod 3)
1 (mod 4)
4 (mod 5)
3 (mod 8)
4 (mod 9)
8 (mod 10)
6 (mod 12)
10 (mod 15)
7 (mod 16)
16 (mod 18)
12 (mod 20)
12 (mod 24)
16 (mod 25)
1 (mod 25)
0 (mod 30)
10 (mod 36)
27 (mod 36)
16 (mod 40)
1 (mod 45)
33 (mod 45)
15 (mod 48)
31 (mod 48)
where 3,4,5,...,48 are the multiplicative orders of 2 modulo the primes 7, 5, 31, 17, 73, 11, 13, 151, 257, 19, 41, 241, 1801, 601, 331, 109, 37, 61681, 23311, 631, 673, 97 respectively.
Now 7 | n*2^k-3 for k == 2 (mod 3) if n == 6 (mod 7),
5 | n*2^k-3 for k == 1 (mod 4) if n == 4 (mod 5), ...,
97 | n*2^k-3 for k == 31 (mod 48) if n == 75 (mod 97).
Using the Chinese remainder theorem, we get infinitely many n for which all these congruences hold, and thus for which n*2^k-3 is always divisible by at least one of those 22 primes.
One such n is 72726958979572419805016319140106929109473069209 (which is not divisible by 3). (End)
For the record high values in this sequence, see A316493; for the indices at which those values occur, see A318561. - Jon E. Schoenfield, Aug 26 2018
Conjecture: For every odd prime p, there exist infinitely many numbers j that are non-multiples of p and have the property that j*2^k - p is composite for every k > 0. - Martin Michael Musatov, Sep 04 2018
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LINKS
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Robert Israel, Table of n, a(n) for n = 1..10000
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MAPLE
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f:= proc(n) local k;
if n mod 3 = 0 then return 0 fi;
for k from 1 do if isprime(n*2^k-3) then return k fi od
end proc:
f(3):= 1:
map(f, [$1..100]); # Robert Israel, Sep 03 2018
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MATHEMATICA
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Array[If[And[Mod[#, 3] == 0, # > 3], 0, Block[{k = 1}, While[! PrimeQ[# 2^k - 3], k++]; k]] &, 105] (* Michael De Vlieger, Sep 04 2018 *)
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PROG
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(PARI) a(n)={my(k=0); if(n%3||n==3, k++; while(!isprime((n<<k)-3), k++)); k} \\ Andrew Howroyd, Aug 24 2018
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CROSSREFS
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Cf. A000040, A050412, A078680.
Sequence in context: A107889 A138384 A129172 * A251482 A172083 A337199
Adjacent sequences: A318288 A318289 A318290 * A318292 A318293 A318294
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KEYWORD
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nonn
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AUTHOR
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Martin Michael Musatov, Aug 23 2018
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EXTENSIONS
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a(3) corrected and a(19)-a(87) from Andrew Howroyd, Aug 25 2018
a(47), a(62), and a(86) corrected by Jon E. Schoenfield, Aug 29 2018
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STATUS
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approved
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