login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A212957 A(n,k) is the number of moduli m such that the multiplicative order of k mod m equals n; square array A(n,k), n>=1, k>=1, read by antidiagonals. 23
0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 3, 2, 2, 2, 0, 2, 5, 4, 6, 1, 0, 4, 2, 3, 4, 4, 3, 0, 2, 6, 2, 12, 6, 10, 1, 0, 4, 4, 8, 4, 9, 16, 2, 4, 0, 3, 6, 2, 26, 4, 37, 6, 14, 2, 0, 4, 3, 12, 18, 4, 10, 3, 8, 4, 5, 0, 2, 12, 5, 14, 6, 42, 2, 28, 26, 16, 3, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Alois P. Heinz, Antidiagonals n = 1..60

Wikipedia, Multiplicative order

FORMULA

A(n,k) = |{m : multiplicative order of k mod m = n}|.

A(n,k) = Sum_{d|n} mu(n/d)*tau(k^d-1), mu = A008683, tau = A000005.

EXAMPLE

A(4,3) = 6: 3^4 = 81 == 1 (mod m) for m in {5,10,16,20,40,80}.

Square array A(n,k) begins:

  0,  1,  2,  2,  3,  2,  4,  2, ...

  0,  1,  2,  2,  5,  2,  6,  4, ...

  0,  1,  2,  4,  3,  2,  8,  2, ...

  0,  2,  6,  4, 12,  4, 26, 18, ...

  0,  1,  4,  6,  9,  4,  4,  6, ...

  0,  3, 10, 16, 37, 10, 42, 24, ...

  0,  1,  2,  6,  3,  2, 12, 10, ...

  0,  4, 14,  8, 28,  8, 48, 72, ...

MAPLE

with(numtheory):

A:= (n, k)-> add(mobius(n/d)*tau(k^d-1), d=divisors(n)):

seq(seq(A(n, 1+d-n), n=1..d), d=1..15);

MATHEMATICA

a[n_, k_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, k^d - 1], {d, Divisors[n]}]; a[1, 1] = 0; Table[ a[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

CROSSREFS

Columns k=1-10 give: A000004, A059499, A059885, A059886, A059887, A059888, A059889, A059890, A059891, A059892.

Rows n=1-10 give: A000005, A059907, A059908, A059909, A059910, A059911, A218256, A218257, A218258, A218259.

Main diagonal gives A252760.

Cf. A000005, A008683.

Sequence in context: A045832 A287528 A289281 * A035393 A068913 A128306

Adjacent sequences:  A212954 A212955 A212956 * A212958 A212959 A212960

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 01 2012

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 February 20 14:52 EST 2019. Contains 320327 sequences. (Running on oeis4.)