A013603 Difference between 2^n and the nearest prime less than or equal to 2^n.

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

Offset: 1

Author

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

Keywords

• nonn

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.
• Corbin Simpson, 2^255 - 19 and Elliptic Curve Cryptography (MegaFavNumbers), Youtube video (2020).

Crossrefs

Cf. A013597, A014234, A049711, A007917, A151799.

Equivalent sequence for next prime: A092131.

Programs

Maple

seq(2^i-prevprime(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 *)

PARI

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

Extensions

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