A339145 a(n) is 1 if A196202(n) is a local maximum, -1 if A196202(n) is a local minimum and 0 otherwise.

0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 0, 1, 0, -1, 1, 0, -1, 1, -1, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 1, 0, -1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 1, -1, 0, 0, 1, 0, -1, 1, 0, -1, 1, -1, 1, 0

Offset

2

Comments

Clearly, if p = prime(n) is a Wieferich prime (A001220) that is preceded and followed by non-Wieferich primes, then a(n) = -1. Heuristic arguments predict this is the case for all Wieferich primes.

How are the terms with value -1 distributed within this sequence?

Is there a correlation between the distribution of Wieferich primes within A000040 and the distribution of the -1 terms within this sequence?

Links

Table of n, a(n) for n=2..79.

Example

For n = 4: The values of A196202(i) for i = 3, 4, 5, respectively, are 16, 15, 56 and 16 > 15 < 56, meaning 15 is a local minimum and therefore a(4) = -1.

Prog

(PARI) a(n) = my(p=prime(n), v=[precprime(p-1), p, nextprime(p+1)]); v=[lift(Mod(2, v[1]^2)^(v[1]-1)), lift(Mod(2, v[2]^2)^(v[2]-1)), lift(Mod(2, v[3]^2)^(v[3]-1))]; if(v[2] > v[1] && v[2] > v[3], return(1), if(v[2] < v[1] && v[2] < v[3], return(-1), return(0)))

Crossrefs

Cf. A001220, A196202.

Sequence in context: A168046 A168184 A013595 * A368912 A368914 A360122

Adjacent sequences: A339142 A339143 A339144 * A339146 A339147 A339148

Keyword

sign,easy

Author

Felix Fröhlich, Nov 25 2020

Status

approved