A338558 Absolute value q such that tau(p) == q (mod p), where p = prime(n) and tau(i) = A000594(i).

0, 0, 0, 0, 1, 5, 7, 7, 1, 5, 10, 6, 11, 12, 20, 24, 14, 12, 3, 19, 6, 37, 20, 33, 20, 27, 50, 34, 36, 29, 18, 64, 4, 2, 66, 32, 3, 64, 61, 51, 60, 84, 95, 83, 63, 97, 42, 28, 61, 67, 32, 10, 29, 73, 37, 92, 16, 120, 31, 107, 120, 141, 145, 39, 12, 74, 150

Offset: 1

Author

Felix Fröhlich, Dec 21 2020

Keywords

• nonn

Comments

These are essentially the values that can be used to define "near-misses" in a search of terms for A007659, similar to how "near-Wieferich primes", "near-Wilson primes" and "near-Wall-Sun-Sun primes" are defined in searches for Wieferich primes (A001220), Wilson primes (A007540) and Wall-Sun-Sun (Fibonacci-Wieferich) primes.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 (mod p), Journal of Integer Sequences 13 (2010), Article 10.7.4.

Crossrefs

Cf. A001220, A007540, A007659, A295645.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme).

Programs

Mathematica

a[n_] := Module[{p = Prime[n]}, Min[Abs[Mod[RamanujanTau[p], {-p, p}]]]]; Array[a, 100] (* Amiram Eldar, Jan 10 2025 *)

PARI

a(n) = my(p=prime(n)); abs(centerlift(Mod(ramanujantau(p), p)))

Formula

a(n) = 0 iff prime(n) is a term of A007659.