

A013603


Difference between 2^n and the nearest prime less than or equal to 2^n.


12



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

1,4


COMMENTS

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


LINKS

T. D. Noe, Table of n, a(n) for n = 1..5000 (corrected by Sean A. Irvine, Jan 18 2019)
V. Danilov, Table for large n [broken link]
Corbin Simpson, 2^255  19 and Elliptic Curve Cryptography (MegaFavNumbers), video (2020)


FORMULA

a(n) = A049711(2^n).  R. J. Mathar, Nov 28 2016
a(n) = 2^n  prevprime(2^n) = 2^n  prime(primepi(2^n)).  Alonso del Arte, Dec 10 2016


MAPLE

seq(2^iprevprime(2^i), i=2..100);


MATHEMATICA

{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 *)


PROG

(PARI) a(n) = 2^n  precprime(2^n); \\ Michel Marcus, Apr 04 2020


CROSSREFS

Cf. A014234, A049711, A007917, A151799.
Sequence in context: A309891 A317937 A322436 * A157892 A264441 A346241
Adjacent sequences: A013600 A013601 A013602 * A013604 A013605 A013606


KEYWORD

nonn


AUTHOR

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)


EXTENSIONS

Extended to a(1) = 0 by M. F. Hasler, Sep 09 2015


STATUS

approved



