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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			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.
		

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.

Crossrefs

Cf. A007918, A007920, A084695, A151800.

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

SageMath

[next_prime(n) - n for n in range(121)] # G. C. Greubel, May 12 2023

Formula

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

Extensions

Incorrect comment removed by Charles R Greathouse IV, Mar 18 2010

More terms from Robert G. Wilson v, Nov 05 2010