A103876 a(n) = -1/10 (mod prime(n)): A test for divisibility by the n-th prime.

2, 1, 9, 5, 17, 16, 26, 3, 11, 4, 30, 14, 37, 53, 6, 20, 7, 51, 71, 58, 80, 29, 10, 72, 32, 98, 79, 38, 13, 41, 125, 134, 15, 47, 114, 50, 121, 161, 18, 19, 135, 59, 179, 21, 156, 68, 206, 163, 215, 24, 25, 77, 184, 242, 27, 83, 28, 198, 205, 92, 31, 219, 95, 33, 101, 104

Offset: 4

Author

Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v, Feb 10 2005

Keywords

• nonn
• base

Comments

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

Values of a(n) can be quickly calculated by finding the smallest multiple of prime(n) which ends in a 1, and removing this last digit. E.g., 7 -> 21 -> 2, 11 -> 11 -> 1, 13 -> 91 -> 9, 17 -> 51 -> 5, 19 -> 171 -> 17.

a(n) is the canonical representative, in the interval (0, p), of the inverse of -10, modulo p = prime(n). - M. F. Hasler, Feb 03 2025

References

• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 76-81.

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..20000

Crossrefs

Cf. A078606, A357913.

Programs

Maple

a:= n-> -1/10 mod ithprime(n):
seq(a(n), n=4..69);  # Alois P. Heinz, Feb 03 2025

Mathematica

a[n_] := Block[{p = Prime[n], k = 1}, While[ Mod[10k + 1, p] != 0, k++ ]; k]; Table[ a[n], {n, 4, 69}]
PowerMod[-10, -1, Prime[Range[4, 100]]] (* Paolo Xausa, Feb 06 2025 *)

PARI

vector(66,n, my(p=prime(n+3)); p-lift(Mod(10,p)^-1)) \\ Joerg Arndt, Jan 23 2023

PARI

a(n)=lift(-1/Mod(10,prime(n)));
apply(a, [4..66]) \\ M. F. Hasler, Feb 03 2025

Python

import sympy
[pow(-10, -1, p) for p in sympy.primerange(7,300)]
# Nicholas Stefan Georgescu, Jan 17 2023

Formula

a(n) = p - (10 mod p)^(-1) where p = prime(n). - Joerg Arndt, Jan 23 2023

Extensions

Definition edited by M. F. Hasler, Feb 03 2025