A357913 - OEIS

%I #54 Feb 07 2025 15:55:04

%S 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,

%T 11,34,89,118,96,14,15,136,110,49,117,52,18,163,172,58,138,20,190,67,

%U 159,23,70,24,217,226,180,79,27,244,194,253,85,88,215,280,94,222,298,236,243

%N Inverse of 10 modulo prime(n).

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

%C 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.

%C 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.

%C Equivalent definition: a(n) = 10^(p - 2) mod p, where p = prime(n). - _Mauro Fiorentini_, Feb 06 2025

%H Paolo Xausa, <a href="/A357913/b357913.txt">Table of n, a(n) for n = 4..10000</a>

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

%F a(n) = (A114013(n) + 1)/10. - _Hugo Pfoertner_, Jan 28 2023

%t PowerMod[10, -1, Prime[Range[4, 100]]] (* _Paolo Xausa_, Feb 07 2025 *)

%o (Python)

%o import sympy

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

%o (PARI) apply( {A357913(n)=lift(1/Mod(10,prime(n)))}, [4..49]) \\ _M. F. Hasler_, Feb 03 2025

%Y Cf. A103876, A078606, A114013.

%K nonn,base

%O 4,1

%A _Nicholas Stefan Georgescu_, Jan 18 2023

%E Better definition from _M. F. Hasler_, Feb 03 2025