Search: a001772 -id:a001772
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A238797
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Smallest k such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime, k <= 2*n+1, or 0 if no such k exists.
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+10
3
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0, 3, 4, 0, 0, 0, 0, 5, 6, 5, 7, 6, 9, 5, 0, 7, 6, 6, 0, 0, 10, 0, 6, 0, 7, 9, 6, 7, 8, 0, 17, 8, 0, 0, 7, 0, 0, 18, 0, 0, 0, 8, 0, 10, 8, 9, 18, 0, 0, 7, 0, 0, 8, 12, 0, 7, 0, 11, 16, 0, 21, 0, 0, 0, 8, 14, 0, 0, 18, 9, 10, 8, 77, 0, 0, 0, 12, 8, 0, 11, 18, 0
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OFFSET
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0,2
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COMMENTS
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Numbers n such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime:
For k = 0: 2, 3, 5, 7, 13, 17, ... Intersection of A000043 and A000043
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...
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LINKS
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EXAMPLE
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a(1) = 3 because 2^3 - (2*1+1) = 5 and (2*1+1)*2^3 - 1 = 23 are both prime, 3 = 2*1+1,
a(2) = 4 because 2^4 - (2*2+1) = 11 and (2*2+1)*2^4 - 1 = 79 are both prime, 4 < 2*2+1 = 5.
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MATHEMATICA
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a[n_] := Catch@ Block[{k = 1}, While[k <= 2*n+1, If[2^k - (2*n + 1) > 0 && PrimeQ[2^k - (2*n+1)] && PrimeQ[(2*n + 1)*2^k-1], Throw@k]; k++]; 0]; a/@ Range[0, 80] (* Giovanni Resta, Mar 15 2014 *)
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CROSSREFS
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Cf. A238748, A238904 (smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k exists).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A050525
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Primes of form 11*2^n-1.
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+10
1
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43, 738197503, 12384898975268863, 198158383604301823, 935776509032580774524280170437362581503, 239558786312340678278215723631964820865023, 1243860333603982568026641440523014635142548663400435746517610765710004322303
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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Select[11 2^Range[0, 250]-1, PrimeQ] (* Harvey P. Dale, Mar 30 2011 *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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A238749
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Exponents of third Mersenne prime pair: numbers n such that 2^n - 5 and 5*2^n - 1 are both prime.
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+10
1
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OFFSET
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1,1
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COMMENTS
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a(7) > 350028.
Exponents of Mersenne prime pairs {2^n - (2k + 1), (2k + 1)*2^n - 1}:
for k = 0: 2, 3, 5, 7, 13, 17, ... Intersection of A000043 and A000043
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...
for k = 14:
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LINKS
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EXAMPLE
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a(1) = 4 because 2^4 - 5 = 11 and 5*2^4 - 1 = 79 are both primes.
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MATHEMATICA
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fQ[n_] := PrimeQ[2^n - 5] && PrimeQ[5*2^n - 1]; k = 1; While[ k < 15001, If[fQ@ k, Print@ k]; k++] (* Robert G. Wilson v, Mar 05 2014 *)
Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5 2^# - 1] &] (* Vincenzo Librandi, May 17 2015 *)
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PROG
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(PARI) isok(n) = isprime(2^n - 5) && isprime(5*2^n - 1); \\ Michel Marcus, Mar 04 2014
(Magma) [n: n in [0..100] | IsPrime(2^n-5) and IsPrime(5*2^n-1)]; // Vincenzo Librandi, May 17 2015
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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