cp's OEIS Frontend

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

%I A162203 #17 Mar 02 2023 16:52:50
%S A162203 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 A162203 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 A162203 -2,1,6,1,-4,1,4,1,-4,1,5,1,-3,1,6,1,-2,1,6
%N A162203 The mountain path of the primes (see comment lines for definition).
%C A162203 On the infinite square grid we draw an infinite straight line from the point (1,0) in direction (2,1).
%C A162203 We start at stage 1 from the point (0,0) drawing an edge ((0,0),(2,0)) in a horizontal direction.
%C A162203 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 A162203 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 A162203 And so on (see illustrations).
%C A162203 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 A162203 The position of the x-th odd noncomposite number A006005(x) is represented by the point P(x,x-1).
%C A162203 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 A162203 See also A162200, A162201 and A162202 for more information.
%H A162203 Antti Karttunen, <a href="/A162203/b162203.txt">Table of n, a(n) for n = 1..20000</a>
%H A162203 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polmpfpn.jpg"> Graph of the mountain path function for prime numbers</a>
%H A162203 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polmpca1.jpg"> Illustration: The mountain path of the primes</a>
%F A162203 From _Nathaniel Johnston_, May 10 2011: (Start)
%F A162203 a(2n+1) = 1 for n >= 2.
%F A162203 a(2n) = (-1)^n*(A162341(n+2) - 1) = (-1)^n*(A052288(n) - 1) + 1 for n >= 2. (End)
%e A162203 Array begins:
%e A162203 =====
%e A162203 X..Y
%e A162203 =====
%e A162203 2, 2;
%e A162203 2, 3;
%e A162203 1,-1;
%e A162203 1, 3;
%e A162203 1,-1;
%e A162203 1, 3;
%e A162203 1,-3;
%e A162203 1, 4;
%e A162203 1,-2;
%e A162203 1, 5;
%o A162203 (PARI)
%o A162203 \\ (After Nathaniel Johnston_'s formula):
%o A162203 A052288(n) = ((prime(n+3) - prime(n+1))/2);
%o A162203 A162203(n) = if(n<=3, 2, if(n%2, 1, 1+((-1)^(n/2)*(A052288(n/2)-1)))); \\ _Antti Karttunen_, Mar 02 2023
%Y A162203 Cf. A000040, A006005, A008578, A162200, A162201, A162202, A162340, A162341, A162342, A162343, A162344.
%K A162203 easy,sign
%O A162203 1,1
%A A162203 _Omar E. Pol_, Jun 27 2009
%E A162203 Edited by _Omar E. Pol_, Jul 02 2009
%E A162203 More terms from _Nathaniel Johnston_, May 10 2011