A357913 -id:A357913 - cp's OEIS frontend

A078606 Constant c(p) used in determining divisibility by the n-th prime, p=A000040(n), for n>=4.

-2, -1, 4, -5, 2, 7, 3, -3, -11, -4, 13, -14, 16, 6, -6, -20, -7, 22, 8, 25, 9, -29, -10, 31, -32, 11, 34, -38, -13, -41, 14, 15, -15, -47, 49, -50, 52, 18, -18, -19, 58, -59, 20, -21, 67, -68, 23, 70, 24, -24, -25, -77, 79, 27, -27, -83, -28, 85, 88, -92, -31, 94, -95, -33, -101, -104, 35, 106, 36, -110, 112, 38

Offset: 4

Devora Rahav (rahav-d(AT)inter.net.il), Dec 09 2002

To determine if N is divisible by p: Write N=10*M+D, where D=ones digit of N and M=N without ones digit. Then N is divisible by p iff M+c(p)*D is.
c(p) is the multiplicative inverse of 10 (mod p) with the smallest absolute value. (But note that the link below uses c(13)=9 rather than -4.)
For a given n, a(n) is either -A103876(n) or prime(n)-A103876(n), whichever has a smaller absolute value. - Nicholas Stefan Georgescu, Jan 18 2023

			The first few terms are c(7)=-2, c(11)=-1, c(13)=4. To find out if a number is divisible by 7, take the last digit, double it and subtract the result from the rest of the number. If you get an answer divisible by 7 (including 0), then the original number is divisible by 7. If you do not know the new number's divisibility, you can apply the rule again. Example: If you had 203, you would subtract 2*3=6 from 20 to get 14; since 14 is divisible by 7, so is 203.

Ethan Magness, Steven Sinnott and Robert L. Ward, Divisibility Rules

Cf. A103876, A357913.

c[p_] := If[(v=PowerMod[10, -1, p])>p/2, v-p, v]; c/@Prime/@Range[4, 100]

a(n) = centerlift(Mod(1,prime(n))/10); \\ Kevin Ryde, Feb 18 2023

import sympy
[(pow(10, -1, p))-p*(p%10%6==1) for p in sympy.primerange(7, 300)]
# Nicholas Stefan Georgescu, Jan 18 2023

Edited by Dean Hickerson, Dec 23 2002

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

Original entry on oeis.org

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

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A078606 Constant c(p) used in determining divisibility by the n-th prime, p=A000040(n), for n>=4.

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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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