A110663 Triangle read by rows: T(n,k) = Sum_{j=k..n} phi(j) (1<=k<=n), where phi is Euler's totient function.

1, 2, 1, 4, 3, 2, 6, 5, 4, 2, 10, 9, 8, 6, 4, 12, 11, 10, 8, 6, 2, 18, 17, 16, 14, 12, 8, 6, 22, 21, 20, 18, 16, 12, 10, 4, 28, 27, 26, 24, 22, 18, 16, 10, 6, 32, 31, 30, 28, 26, 22, 20, 14, 10, 4, 42, 41, 40, 38, 36, 32, 30, 24, 20, 14, 10, 46, 45, 44, 42, 40, 36, 34, 28, 24, 18, 14, 4

Offset: 1

Author

Emeric Deutsch, Aug 02 2005

Keywords

• nonn
• tabl

Examples

			T(5,3) = 8 because phi(3)+phi(4)+phi(5) = 2+2+4 = 8.
Triangle begins:
1;
2,1;
4,3,2;
6,5,4,2;
10,9,8,6,4;
...
		

Links

• Indranil Ghosh, Rows 1..100, flattened

Crossrefs

Cf. A000010, A002088.

Programs

Mathematica

T[n_, n_] := EulerPhi[n]; T[n_, k_] := Sum[EulerPhi[j], {j, k, n}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Sep 03 2017 *)

PARI

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(sum(j=k, n, eulerphi(j)), ", ");); print(););} \\ Michel Marcus, Apr 05 2015

Formula

T(n,n) = phi(n) = A000010(n) = number of numbers <=n and relatively prime to n.

T(n,1) = Sum_{j=1..n} phi(j) = A002088(n).