A143230 Triangle read by rows, A130207 * A000012 * A130207.

1, 1, 1, 2, 2, 4, 2, 2, 4, 4, 4, 4, 8, 8, 16, 2, 2, 4, 4, 8, 4, 6, 6, 12, 12, 24, 12, 36, 4, 4, 8, 8, 16, 8, 24, 16, 6, 6, 12, 12, 24, 12, 36, 24, 36, 4, 4, 8, 8, 16, 8, 24, 16, 24, 16, 10, 10, 20, 20, 40, 20, 60, 40, 60, 40, 100, 4, 4, 8, 8, 16, 8, 24, 16, 24, 16, 40, 16

Offset

1,4

Comments

Left border = A000010.

Row sums = A143231: (1, 2, 8, 12, 40, 24, ...).

T(n,k) is the number of pairs (a,b), where 0 <= a < n, 0 <= b < k, gcd(a,n) != 1, and gcd(b,k) != 1. - Joerg Arndt, Jun 26 2011

Links

Nathaniel Johnston, Rows 1..100, flattened

Formula

Triangle read by rows, A130207 * A000012 * A130207, where A130207 = A000010 * 0^(n-k), 1 <= k <= n.

T(n,k) = phi(n) * phi(k), where phi(n) & phi(k) = Euler's totient function.

Example

First few rows of the triangle:
  1;
  1,  1;
  2,  2,  4;
  2,  2,  4,  4;
  4,  4,  8,  8, 16;
  2,  2,  4,  4,  8,  4;
  6,  6, 12, 12, 24, 12, 36;
  4,  4,  8,  8, 16,  8, 24, 16;
  6,  6, 12, 12, 24, 12, 36, 24, 36;
  ...
T(7,5) = 24 = phi(7) * phi(5) = 6 * 4.

Maple

with(numtheory): T := proc(n, k) return phi(n)*phi(k): end: seq(seq(T(n, k), k=1..n), n=1..12); # Nathaniel Johnston, Jun 26 2011

Crossrefs

Cf. A000010, A130207, A143231.

Sequence in context: A093621 A242734 A309709 * A276604 A072301 A127171

Adjacent sequences: A143227 A143228 A143229 * A143231 A143232 A143233

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Jul 31 2008

Status

approved