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

1,1

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.

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

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)

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A307298 A216674 A106795 * A071455 A288724 A198862

Adjacent sequences: A162200 A162201 A162202 * A162204 A162205 A162206

Keyword

easy,sign

Author

Omar E. Pol, Jun 27 2009

Extensions

Edited by Omar E. Pol, Jul 02 2009

More terms from Nathaniel Johnston, May 10 2011

Status

approved