A078606 -id:A078606 - cp's OEIS frontend

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

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

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

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

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

Cf. A078606, A357913.

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

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 *)

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

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

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

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

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

A357913 Inverse of 10 modulo prime(n).

Original entry on oeis.org

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

Nicholas Stefan Georgescu, Jan 18 2023

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

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

Cf. A103876, A078606, A114013.

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

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

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

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

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

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

cp's OEIS Frontend

A078149 See A078606 for the official version of this sequence.

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

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A357913 Inverse of 10 modulo prime(n).

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Mathematica

PARI

Python

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