A054522 Triangle T(n,k): T(n,k) = phi(k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1<=k<=n). T(n,k) = number of elements of order k in cyclic group of order n.

1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 1, 0, 0, 0, 4, 1, 1, 2, 0, 0, 2, 1, 0, 0, 0, 0, 0, 6, 1, 1, 0, 2, 0, 0, 0, 4, 1, 0, 2, 0, 0, 0, 0, 0, 6, 1, 1, 0, 0, 4, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 1, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0

Offset

1,6

Comments

T(n,1) = 1; T(n,n) = A000010(n).

This triangle is the transpose of the upper triangular array U in the LU decomposition of the square array A003989. - Peter Bala, Oct 15 2023

Links

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened

R. J. Mathar, Plots of cycle graphs of the finite groups up to order 36, (2015)

Formula

Sum (T(n,k): k = 1 .. n) = n. - Reinhard Zumkeller, Oct 18 2011

Example

1;
1, 1;
1, 0, 2;
1, 1, 0, 2;
1, 0, 0, 0, 4;
1, 1, 2, 0, 0, 2;
1, 0, 0, 0, 0, 0, 6;
1, 1, 0, 2, 0, 0, 0, 4;
1, 0, 2, 0, 0, 0, 0, 0, 6;

Maple

A054522 := proc(n, k)
    if modp(n, k) = 0 then
        numtheory[phi](k) ;
    else
        0;
    end if;
end proc:
seq(seq(A054522(n, k), k=1..n), n=1..15) ; # R. J. Mathar, Aug 06 2016

Mathematica

t[n_, k_] /; Divisible[n, k] := EulerPhi[k]; t[_, _] = 0; Flatten[Table[t[n, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, Nov 25 2011 *)
Flatten[Table[If[Divisible[n, k], EulerPhi[k], 0], {n, 15}, {k, n}]] (* Harvey P. Dale, Feb 27 2012 *)

Prog

(PARI) T(n, k)=if(k<1 || k>n, 0, if(n%k, 0, eulerphi(k)))
(Haskell)
a054522 n k = a054522_tabl !! (n-1) !! (k-1)
a054522_tabl = map a054522_row [1..]
a054522_row n = map (\k -> if n `mod` k == 0 then a000010 k else 0) [1..n]
-- Reinhard Zumkeller, Oct 18 2011

Crossrefs

Cf. A003989, A054521.

Sequence in context: A035183 A178101 A324831 * A110250 A065252 A115211

Adjacent sequences: A054519 A054520 A054521 * A054523 A054524 A054525

Keyword

nonn,tabl,nice,easy

Author

N. J. A. Sloane, Apr 09 2000

Status

approved