A162203 The mountain path of the primes (see comment lines for definition).

2, 2, 2, 3, 1, -1, 1, 3, 1, -1, 1, 3, 1, -3, 1, 4, 1, -2, 1, 5, 1, -1, 1, 3, 1, -3, 1, 6, 1, -2, 1, 4, 1, -3, 1, 3, 1, -2, 1, 5, 1, -3, 1, 7, 1, -4, 1, 3, 1, -1, 1, 3, 1, -1, 1, 9, 1, -7, 1, 5, 1, -2, 1, 6, 1, -4, 1, 4, 1, -4, 1, 5, 1, -3, 1, 6, 1, -2, 1, 6

Offset: 1

Author

Omar E. Pol, Jun 27 2009

Keywords

• easy
• sign

Comments

On the infinite square grid we draw an infinite straight line from the point (1,0) in direction (2,1).

We start at stage 1 from the point (0,0) drawing an edge ((0,0),(2,0)) in a horizontal direction.

At stage 2 we draw an edge ((2,0),(2,2)) in a vertical direction. We can see that the straight line intercepts at the number 3 (the first odd prime).

At stage 3 we draw an edge ((2,2),(4,2)) in a horizontal direction. We can see that the straight line intercepts at the number 5 (the second odd prime).

And so on (see illustrations).

The absolute value of a(n) is equal to the length of the n-th edge of a path, or infinite square polyedge, such that the mentioned straight line intercepts, on the path, at the number 1 and the odd primes. In other words, the straight line intercepts the odd noncomposite numbers (A006005).

The position of the x-th odd noncomposite number A006005(x) is represented by the point P(x,x-1).

So the position of the first prime number is represented by the point P(2,0) and position of the x-th prime A000040(x), for x>1, is represented by the point P(x,x-1); for example, 31, the 11th prime, is represented by the point P(11,10).

See also A162200, A162201 and A162202 for more information.

Examples

			Array begins:
=====
X..Y
=====
2, 2;
2, 3;
1,-1;
1, 3;
1,-1;
1, 3;
1,-3;
1, 4;
1,-2;
1, 5;
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Omar E. Pol, Graph of the mountain path function for prime numbers
• Omar E. Pol, Illustration: The mountain path of the primes

Crossrefs

Cf. A000040, A006005, A008578, A162200, A162201, A162202, A162340, A162341, A162342, A162343, A162344.

Programs

PARI

\\ (After Nathaniel Johnston_'s formula):
A052288(n) = ((prime(n+3) - prime(n+1))/2);
A162203(n) = if(n<=3, 2, if(n%2, 1, 1+((-1)^(n/2)*(A052288(n/2)-1)))); \\ Antti Karttunen, Mar 02 2023

Formula

From Nathaniel Johnston, May 10 2011: (Start)

a(2n+1) = 1 for n >= 2.

a(2n) = (-1)^n*(A162341(n+2) - 1) = (-1)^n*(A052288(n) - 1) + 1 for n >= 2. (End)

Extensions

Edited by Omar E. Pol, Jul 02 2009

More terms from Nathaniel Johnston, May 10 2011