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%I #43 Oct 19 2022 06:37:43
%S 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,
%T 5,9,41,31,5,25,45,7,87,21,11,57,17,55,21,115,59,81,27,129,47,111,33,
%U 55,5,13,27,55,93,1,57,25,59,49,5,19,23,19,35,231,93,69,35,97,15
%N Difference between 2^n and the nearest prime less than or equal to 2^n.
%C If a(n) = 1, then n is prime and 2^n - 1 is a Mersenne prime. - _Franz Vrabec_, Sep 27 2005
%C 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
%C 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
%H T. D. Noe, <a href="/A013603/b013603.txt">Table of n, a(n) for n = 1..5000</a> (corrected by Sean A. Irvine, Jan 18 2019)
%H V. Danilov, <a href="https://web.archive.org/web/20060127010153/http://www.fortunecity.com/skyscraper/epson/276/pr1_2k.htm">Table for large n</a>.
%H Corbin Simpson, <a href="https://www.youtube.com/watch?v=LfepFANkz-k">2^255 - 19 and Elliptic Curve Cryptography (MegaFavNumbers)</a>, Youtube video (2020).
%F a(n) = A049711(2^n). - _R. J. Mathar_, Nov 28 2016
%F a(n) = 2^n - prevprime(2^n) = 2^n - prime(primepi(2^n)). - _Alonso del Arte_, Dec 10 2016
%p seq(2^i-prevprime(2^i),i=2..100);
%t {0} ~Join~ Array[With[{c = 2^#}, c - NextPrime[c, -1]] &, 80, 2] (* _Harvey P. Dale_, Jul 23 2013 *)
%t Table[2^n - Prime[PrimePi[2^n]], {n, 80}] (* _Alonso del Arte_, Dec 10 2016 *)
%o (PARI) a(n) = 2^n - precprime(2^n); \\ _Michel Marcus_, Apr 04 2020
%Y Cf. A013597, A014234, A049711, A007917, A151799.
%Y Equivalent sequence for next prime: A092131.
%K nonn
%O 1,4
%A James Kilfiger (mapdn(AT)csv.warwick.ac.uk)
%E Extended to a(1) = 0 by _M. F. Hasler_, Sep 09 2015
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