A357913 Inverse of 10 modulo prime(n).

5, 10, 4, 12, 2, 7, 3, 28, 26, 37, 13, 33, 16, 6, 55, 47, 64, 22, 8, 25, 9, 68, 91, 31, 75, 11, 34, 89, 118, 96, 14, 15, 136, 110, 49, 117, 52, 18, 163, 172, 58, 138, 20, 190, 67, 159, 23, 70, 24, 217, 226, 180, 79, 27, 244, 194, 253, 85, 88, 215, 280, 94, 222, 298, 236, 243

Offset: 4

Author

Nicholas Stefan Georgescu, Jan 18 2023

Keywords

• nonn
• base

Comments

Original definition: "Another test for divisibility by the n-th prime (see Comments for precise definition)."

Given a number M, delete its last digit d, then add d*a(n). If the result is divisible by prime(n), then M is also divisible by prime(n). This process may be repeated.

a(n) can be quickly calculated by finding the smallest multiple of prime(n) ending in 9, adding one, and dividing that result by 10. E.g., 7 -> 49 -> 5, 11 -> 99 -> 10, 13 -> 39 -> 4, 17 -> 119 -> 12, 19 -> 19 -> 2.

Equivalent definition: a(n) = 10^(p - 2) mod p, where p = prime(n). - Mauro Fiorentini, Feb 06 2025

Links

• Paolo Xausa, Table of n, a(n) for n = 4..10000

Crossrefs

Cf. A103876, A078606, A114013.

Programs

Mathematica

PowerMod[10, -1, Prime[Range[4, 100]]] (* Paolo Xausa, Feb 07 2025 *)

PARI

apply( {A357913(n)=lift(1/Mod(10,prime(n)))}, [4..49]) \\ M. F. Hasler, Feb 03 2025

Python

import sympy
[pow(10, -1, p) for p in sympy.primerange(7,348)]

Formula

a(n) = prime(n) - A103876(n).

a(n) = (A114013(n) + 1)/10. - Hugo Pfoertner, Jan 28 2023

Extensions

Better definition from M. F. Hasler, Feb 03 2025