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A013603
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Difference between 2^n and the nearest prime less than or equal to 2^n.
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16
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0, 1, 1, 3, 1, 3, 1, 5, 3, 3, 9, 3, 1, 3, 19, 15, 1, 5, 1, 3, 9, 3, 15, 3, 39, 5, 39, 57, 3, 35, 1, 5, 9, 41, 31, 5, 25, 45, 7, 87, 21, 11, 57, 17, 55, 21, 115, 59, 81, 27, 129, 47, 111, 33, 55, 5, 13, 27, 55, 93, 1, 57, 25, 59, 49, 5, 19, 23, 19, 35, 231, 93, 69, 35, 97, 15
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OFFSET
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1,4
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COMMENTS
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If a(n) = 1, then n is prime and 2^n - 1 is a Mersenne prime. - Franz Vrabec, Sep 27 2005
Using the first variant A007917 (rather than A151799) of the prevprime() function, the sequence is well defined for n = 1, with a(1) = 2^1 - prevprime(2^1) = 2 - 2 = 0. - M. F. Hasler, Sep 09 2015
In Mathematica, one can use NextPrime with a second argument of -1 to obtain the next smaller prime. As almost all the powers of 2 are composite, this produces the proper results for most of this sequence. However, NextPrime[2, -1] returns -2 rather than the expected 2, which would consequently mean a(1) = 4 rather than 0. - Alonso del Arte, Dec 10 2016
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LINKS
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FORMULA
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a(n) = 2^n - prevprime(2^n) = 2^n - prime(primepi(2^n)). - Alonso del Arte, Dec 10 2016
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MAPLE
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seq(2^i-prevprime(2^i), i=2..100);
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MATHEMATICA
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{0} ~Join~ Array[With[{c = 2^#}, c - NextPrime[c, -1]] &, 80, 2] (* Harvey P. Dale, Jul 23 2013 *)
Table[2^n - Prime[PrimePi[2^n]], {n, 80}] (* Alonso del Arte, Dec 10 2016 *)
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PROG
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(PARI) a(n) = 2^n - precprime(2^n); \\ Michel Marcus, Apr 04 2020
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CROSSREFS
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Equivalent sequence for next prime: A092131.
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KEYWORD
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nonn
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AUTHOR
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James Kilfiger (mapdn(AT)csv.warwick.ac.uk)
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EXTENSIONS
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STATUS
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approved
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