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A278603 A prime mountain: peaks and valleys beyond the origin correspond to prime abscissa (see Comments for precise definition). 1
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, 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, 9, 8, 7, 6, 5, 6, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

We start with a(0)=0 and a(1)=1, and then the sequence is extended according to these rules:

(1) |a(n+1) - a(n)| = 1 for any n>1,

(2) a(n+1) = a(n-1) iff n is prime.

Is this sequence ultimately positive or ultimately negative or will it change sign indefinitely?

From Ryan Bresler, Jan 04 2021: (Start)

This sequence will contain every integer on "at least one side" of the origin, i.e., it will not have a finite range.

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.

(End)

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..10000

FORMULA

a(prime(n)) = prime(1) + Sum_{k=1..n-1} A001223(k)*(-1)^k for any n > 0.

a(n+1) = A065358(n) + 1 for any n >= 0. - Rémy Sigrist, Feb 22 2018

EXAMPLE

a(2) is either a(1) + 1 = 2 or a(1) - 1 = 0.

As 1 is not prime, a(2) = a(1+1) != a(1-1) = 0.

Hence, a(2) = 2.

As 2 is prime, a(3) = a(2+1) = a(2-1) = a(1) = 1.

As 3 is prime, a(4) = a(3+1) = a(3-1) = a(2) = 2.

a(5) is either a(4)+1 = 3 or a(4)-1 = 1.

As 4 is not prime, a(5) = a(4+1) != a(4-1) = 1.

Hence, a(5) = 3.

The first terms can be visualized here (peaks correspond to odd-indexed primes, and valleys to even-indexed primes):

.                  /\  ...

.                 /  \/

.            /\  /

.           /  \/

.      /\  /

.   /\/  \/

.  /

.   2  5     11    17

. 0  3   7     13    19

PROG

(PARI) y=0; slope=+1; for (x=0, 85, print1 (y ", "); if (isprime(x), slope = -slope); y+=slope)

CROSSREFS

Cf. A001223, A065358.

Sequence in context: A005811 A008342 A277214 * A248218 A182110 A175328

Adjacent sequences:  A278600 A278601 A278602 * A278604 A278605 A278606

KEYWORD

sign

AUTHOR

Rémy Sigrist, Nov 23 2016

STATUS

approved

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Last modified April 11 03:01 EDT 2021. Contains 342886 sequences. (Running on oeis4.)