A013632 - OEIS

A013632

Difference between n and the next prime greater than n.

2, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 2, 1, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 2, 1, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3

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OFFSET

0,1

COMMENTS

Alternatively, a(n) is the smallest positive k such that n + k is prime. - N. J. A. Sloane, Nov 18 2015

Except for a(0) and a(1), a(n) is the least k such that gcd(n!, n + k) = 1. - Robert G. Wilson v, Nov 05 2010

This sequence uses the "strictly larger" variant A151800 of the nextprime function, rather than A007918. Therefore all terms are positive and a(n) = 1 if and only if n + 1 is a prime. - M. F. Hasler, Sep 09 2015

For n > 0, a(n) and n are of opposite parity. Also, by Bertrand's postulate (actually a theorem), for n > 1, a(n) < n. - Zak Seidov, Dec 27 2018

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Brăduţ Apostol, Laurenţiu Panaitopol, Lucian Petrescu, and László Tóth, Some properties of a sequence defined with the aid of prime numbers, arXiv:1503.01086 [math.NT], 2015.

Brăduţ Apostol, Laurenţiu Panaitopol, Lucian Petrescu, and László Tóth, Some Properties of a Sequence Defined with the Aid of Prime Numbers, J. Int. Seq. 18 (2015) # 15.5.5.

FORMULA

a(n) = Prime(1 + PrimePi(n)) - n = A084695(n, 1) (for n > 0). - G. C. Greubel, May 12 2023

EXAMPLE

a(30) = 1 because 31 is the next prime greater than 30 and 31 - 30 = 1.

a(31) = 6 because 37 is the next prime greater than 31 and 37 - 31 = 6.

MAPLE

[ seq(nextprime(i)-i, i=0..100) ];

MATHEMATICA

Array[NextPrime[#] - # &, 105, 0] (* Robert G. Wilson v, Nov 05 2010 *)

PROG

(PARI) a(n) = nextprime(n+1) - n; \\ Michel Marcus, Mar 04 2015

(Magma) [NextPrime(n) - n: n in [0..100]]; // Vincenzo Librandi, Dec 27 2018