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

Author

N. J. A. Sloane, Apr 09 2000

Keywords

• nonn
• tabl
• nice
• easy

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

Examples

			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;
		

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)

Crossrefs

Cf. A003989, A054521, A008683, A050873.

Programs

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

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 *)

PARI

T(n,k)=if(k<1 || k>n,0,if(n%k,0,eulerphi(k)))

Formula

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

T(n,k) = Sum_{d|k} mu(k/d)*gcd(n,d). - Ridouane Oudra, Apr 05 2025