login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A102057 Triangle of modular inverses of b (mod m) for b = 1, ..., m-1, where 0 indicates no modular inverse exists. 4
1, 1, 2, 1, 0, 3, 1, 3, 2, 4, 1, 0, 0, 0, 5, 1, 4, 5, 2, 3, 6, 1, 0, 3, 0, 5, 0, 7, 1, 5, 0, 7, 2, 0, 4, 8, 1, 0, 7, 0, 0, 0, 3, 0, 9, 1, 6, 4, 3, 9, 2, 8, 7, 5, 10, 1, 0, 0, 0, 5, 0, 7, 0, 0, 0, 11, 1, 7, 9, 10, 8, 11, 2, 5, 3, 4, 6, 12, 1, 0, 5, 0, 3, 0, 0, 0, 11, 0, 9, 0, 13, 1, 8, 0, 4, 0, 0, 13 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For all m, a(m, 1) = 1 and a(m, m-1) = m-1. - Evgeny Kapun, Dec 20 2016

For all coprime m and b, a(m, b)*b == 1 (mod m) and a(m, a(m, b)) = b. - Evgeny Kapun, Dec 20 2016

LINKS

Evgeny Kapun, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Modular Inverse

FORMULA

From Evgeny Kapun, Dec 20 2016: (Start)

If gcd(m, b) = 1 = mx+by (Bézout's lemma), then a(m, b) == y (mod m).

If gcd(m, b) = 1, then a(m, b) == b^(phi(m)-1) == b^(psi(m)-1) (mod m), where phi(n) is A000010 and psi(n) is A002322.

If m is prime, then a(m, b) == b^(m-2) (mod m).

If m is prime, then a(m, 1) = 1 and a(m, b) == -floor(m/b)a(m, m mod b) (mod m).

(End)

EXAMPLE

m\b 1 2 3 4 5 6 7 8 9 10 11

2   1

3   1 2

4   1 0 3

5   1 3 2 4

6   1 0 0 0 5

7   1 4 5 2 3 6

8   1 0 3 0 5 0 7

9   1 5 0 7 2 0 4 8

10  1 0 7 0 0 0 3 0 9

11  1 6 4 3 9 2 8 7 5 10

12  1 0 0 0 5 0 7 0 0  0 11

MATHEMATICA

Table[ If[ !CoprimeQ[b, m], 0, PowerMod[b, -1, m]], {m, 1, 15}, {b, 1, m-1}] // Flatten (* Jean-François Alcover, Nov 09 2012, after Eric W. Weisstein *)

PROG

(Haskell) [let g 1 0 x y = x; g a 0 x y = 0; g a b x y = g b (mod a b) y (x - y * div a b) in g m b 0 1 `mod` m | m <- [1..], b <- [1..m-1]] -- Evgeny Kapun, Dec 20 2016

CROSSREFS

Sequence in context: A146540 A162922 A246180 * A276276 A157497 A257961

Adjacent sequences:  A102054 A102055 A102056 * A102058 A102059 A102060

KEYWORD

nonn,tabl

AUTHOR

Eric W. Weisstein, Dec 28 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 28 05:55 EST 2020. Contains 338699 sequences. (Running on oeis4.)