A162203 - OEIS

%I #17 Mar 02 2023 16:52:50

%S 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,

%T 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,

%U -2,1,6,1,-4,1,4,1,-4,1,5,1,-3,1,6,1,-2,1,6

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

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

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

%C 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).

%C 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).

%C And so on (see illustrations).

%C 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).

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

%C 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).

%C See also A162200, A162201 and A162202 for more information.

%H Antti Karttunen, <a href="/A162203/b162203.txt">Table of n, a(n) for n = 1..20000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polmpfpn.jpg"> Graph of the mountain path function for prime numbers</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polmpca1.jpg"> Illustration: The mountain path of the primes</a>

%F From _Nathaniel Johnston_, May 10 2011: (Start)

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

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

%e Array begins:

%e =====

%e X..Y

%e =====

%e 2, 2;

%e 2, 3;

%e 1,-1;

%e 1, 3;

%e 1,-1;

%e 1, 3;

%e 1,-3;

%e 1, 4;

%e 1,-2;

%e 1, 5;

%o (PARI)

%o \\ (After Nathaniel Johnston_'s formula):

%o A052288(n) = ((prime(n+3) - prime(n+1))/2);

%o A162203(n) = if(n<=3, 2, if(n%2, 1, 1+((-1)^(n/2)*(A052288(n/2)-1)))); \\ _Antti Karttunen_, Mar 02 2023

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

%K easy,sign

%O 1,1

%A _Omar E. Pol_, Jun 27 2009

%E Edited by _Omar E. Pol_, Jul 02 2009

%E More terms from _Nathaniel Johnston_, May 10 2011