A162345 Length of n-th edge in the graph of the zig-zag function for prime numbers.

2, 2, 2, 3, 3, 3, 3, 3, 5, 4, 4, 5, 3, 3, 5, 6, 4, 4, 5, 3, 4, 5, 5, 7, 6, 3, 3, 3, 3, 9, 9, 5, 4, 6, 6, 4, 6, 5, 5, 6, 4, 6, 6, 3, 3, 7, 12, 8, 3, 3, 5, 4, 6, 8, 6, 6, 4, 4, 5, 3, 6, 12, 9, 3, 3, 9, 10, 8, 6, 3, 5, 7, 7, 6, 5, 5, 7, 6, 6, 9, 6, 6, 6, 4, 5, 5

Offset: 1

Author

Omar E. Pol, Jul 04 2009

Keywords

• easy
• nonn

Comments

Also, first differences of A162800.

Also {2, 2, } together with the numbers A052288.

Note that the graph of the zig-zag function for prime numbers is similar to the graph of the mountain path function for prime numbers but with exactly a vertex between consecutive odd noncomposite numbers (A006005).

This is the same as A115061 if n>1 (and also essentially equal to A052288). Proof: Because this is the first differences of A162800, which is {0,2} together with A024675, this sequence (for n>=3) is given by a(n) = (prime(n+1) - prime(n-1))/2. Similarly, because half the numbers between prime(n-1) and prime(n+1) are closer to prime(n) than any other prime, A115061(n) = (prime(n+1) - prime(n-1))/2 for n>=3 as well. - Nathaniel Johnston, Jun 25 2011

Examples

			Array begins:
=====
x, y
=====
2, 2;
2, 3;
3, 3;
3, 3;
5, 4;
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• Omar E. Pol, Graph of the mountain path function for prime numbers

Crossrefs

Cf. A000040, A006005, A008578, A024675, A052288, A162203, A162800, A162801, A162802.

Programs

Magma

[2,2] cat[(NthPrime(n+1)-NthPrime(n-1))/2: n in [3..80]]; // Vincenzo Librandi, Dec 19 2016

Maple

A162345 := proc(n) if(n<=2)then return 2: fi: return (ithprime(n+1) - ithprime(n-1))/2: end: seq(A162345(n),n=1..100); # Nathaniel Johnston, Jun 25 2011

Mathematica

Join[{2, 2}, Table[(Prime[n+1] - Prime[n-1])/2, {n, 3, 100}]] (* Vincenzo Librandi, Dec 19 2016 *)

Formula

a(n) = (prime(n+1) - prime(n-1))/2 for n>=3. - Nathaniel Johnston, Jun 25 2011

Extensions

Edited by Omar E. Pol, Jul 16 2009