OFFSET
4,1
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.
REFERENCES
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 76-81.
FORMULA
a(n) = p - (10 mod p)^(-1) where p = prime(n). - Joerg Arndt, Jan 23 2023
MATHEMATICA
a[n_] := Block[{p = Prime[n], k = 1}, While[ Mod[10k + 1, p] != 0, k++ ]; k]; Table[ a[n], {n, 4, 69}]
PROG
(Python)
import sympy
[pow(-10, -1, p) for p in sympy.primerange(7, 300)]
# Nicholas Stefan Georgescu, Jan 17 2023
(PARI) vector(66, n, my(p=prime(n+3)); p-lift(Mod(10, p)^-1)) \\ Joerg Arndt, Jan 23 2023
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v, Feb 10 2005
STATUS
approved