A103876 -id:A103876 - 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

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

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

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