A046933 Number of composites between successive primes.

0, 1, 1, 3, 1, 3, 1, 3, 5, 1, 5, 3, 1, 3, 5, 5, 1, 5, 3, 1, 5, 3, 5, 7, 3, 1, 3, 1, 3, 13, 3, 5, 1, 9, 1, 5, 5, 3, 5, 5, 1, 9, 1, 3, 1, 11, 11, 3, 1, 3, 5, 1, 9, 5, 5, 5, 1, 5, 3, 1, 9, 13, 3, 1, 3, 13, 5, 9, 1, 3, 5, 7, 5, 5, 3, 5, 7, 3, 7, 9, 1, 9, 1, 5, 3, 5, 7, 3, 1, 3, 11, 7, 3, 7, 3, 5, 11, 1, 17

Offset: 1

Author

Marc LeBrun, Dec 11 1999

Keywords

• easy
• nonn
• nice

Comments

a(n) is odd for n>1 since all primes except 2 are odd. - Joel Brennan, Jan 02 2023

Examples

			a(1) = 0 since 2 is adjacent to 3;
a(2) = 1 since 4 is between 3 and 5;
a(4) = 3 = 11 - 7 - 1, etc.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A028334, A036263.

Cf. A008996 (record values > 0).

Programs

Haskell

a046933 n = a046933_list !! (n-1)
a046933_list = map (subtract 1) a001223_list
-- Reinhard Zumkeller, Dec 12 2012

Maple

A046933:=n->ithprime(n+1)-ithprime(n)-1; seq(A046933(n), n=1..100); # Wesley Ivan Hurt, Apr 15 2014

Mathematica

Differences[Prime[Range[100]]] - 1 (* Arkadiusz Wesolowski, Nov 18 2011 *)
Table[Prime[n + 1] - Prime[n] - 1, {n, 100}] (* Wesley Ivan Hurt, Apr 15 2014 *)
Prepend[Drop[Length/@SequenceSplit[Range@Prime@100,{?PrimeQ}],1],0] (* _Federico Provvedi, Jul 19 2021 *)

PARI

a(n)=prime(n+1)-prime(n)-1 \\ Charles R Greathouse IV, Nov 20 2012

Python

from sympy import prime
def A046933(n): return prime(n+1)-prime(n)-1 # Chai Wah Wu, Jan 02 2024

Formula

a(n) = prime(n+1) - prime(n) - 1 = A000040(n+1) - A000040(n) - 1.

a(n) = A001223(n) - 1.

a(n) = 2*A028334(n) - 1 for n>1. - Giovanni Teofilatto, Apr 19 2010

a(n) = Sum_{i=1..n-1} A036263(i). - Daniel Forgues, Apr 07 2014