|
%I #42 Jan 09 2021 02:46:53
%S 0,1,2,1,2,3,2,1,2,3,4,5,4,3,4,5,6,7,6,5,6,7,8,9,8,7,6,5,4,3,4,5,4,3,
%T 2,1,0,-1,0,1,2,3,2,1,2,3,4,5,4,3,2,1,0,-1,0,1,2,3,4,5,4,3,4,5,6,7,8,
%U 9,8,7,6,5,6,7,6,5,4,3,2,1,2,3,4,5,4,3
%N A prime mountain: peaks and valleys beyond the origin correspond to prime abscissa (see Comments for precise definition).
%C We start with a(0)=0 and a(1)=1, and then the sequence is extended according to these rules:
%C (1) |a(n+1) - a(n)| = 1 for any n>1,
%C (2) a(n+1) = a(n-1) iff n is prime.
%C Is this sequence ultimately positive or ultimately negative or will it change sign indefinitely?
%C From _Ryan Bresler_, Jan 04 2021: (Start)
%C This sequence will contain every integer on "at least one side" of the origin, i.e., it will not have a finite range.
%C Suppose this sequence has both a finite minimum, R1, and a finite maximum, R2. Since prime gaps become arbitrarily large, we will eventually reach a prime gap g, such that g > R2 - R1. We can see that this prime gap will cause at least one term of this sequence to be outside the interval [R1, R2]. This contradiction shows that all integers on at least one side of the origin will be terms of the sequence.
%C (End)
%H Rémy Sigrist, <a href="/A278603/b278603.txt">Table of n, a(n) for n = 0..10000</a>
%F a(prime(n)) = prime(1) + Sum_{k=1..n-1} A001223(k)*(-1)^k for any n > 0.
%F a(n+1) = A065358(n) + 1 for any n >= 0. - _Rémy Sigrist_, Feb 22 2018
%e a(2) is either a(1) + 1 = 2 or a(1) - 1 = 0.
%e As 1 is not prime, a(2) = a(1+1) != a(1-1) = 0.
%e Hence, a(2) = 2.
%e As 2 is prime, a(3) = a(2+1) = a(2-1) = a(1) = 1.
%e As 3 is prime, a(4) = a(3+1) = a(3-1) = a(2) = 2.
%e a(5) is either a(4)+1 = 3 or a(4)-1 = 1.
%e As 4 is not prime, a(5) = a(4+1) != a(4-1) = 1.
%e Hence, a(5) = 3.
%e The first terms can be visualized here (peaks correspond to odd-indexed primes, and valleys to even-indexed primes):
%e . /\ ...
%e . / \/
%e . /\ /
%e . / \/
%e . /\ /
%e . /\/ \/
%e . /
%e . 2 5 11 17
%e . 0 3 7 13 19
%o (PARI) y=0; slope=+1; for (x=0, 85, print1 (y ", "); if (isprime(x), slope = -slope); y+=slope)
%Y Cf. A001223, A065358.
%K sign
%O 0,3
%A _Rémy Sigrist_, Nov 23 2016
|
