A224777 Triangle with integer geometric mean sqrt(n*m) for 1 <= m <= n, and 0 if sqrt(n*m) is not integer.

1, 0, 2, 0, 0, 3, 2, 0, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 7, 0, 4, 0, 0, 0, 0, 0, 8, 3, 0, 0, 6, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0

Offset: 1

Author

Wolfdieter Lang, Apr 25 2013

Keywords

• nonn
• tabl

Comments

If the numbers > 1 are replaced by 1 one obtains the corresponding characteristic triangle. a(n,n) = n. a(n,1) = sqrt(n) iff n is a square.

The number of nonzero entries in row n is A000188(n).

For n and m with gcd(n,m) = 1 the nonzero entries are precisely a(N^2,M^2) = N*M, with integers N, M satisfying gcd(N,M) = 1 , 1 <= M <= N. - Wolfdieter Lang, Apr 26 2013

Examples

			The triangle begins:
n\m 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
1:  1
2:  0  2
3:  0  0  3
4:  2  0  0  4
5:  0  0  0  0  5
6:  0  0  0  0  0  6
7:  0  0  0  0  0  0  7
8:  0  4  0  0  0  0  0  8
9:  3  0  0  6  0  0  0  0  9
10: 0  0  0  0  0  0  0  0  0 10
11: 0  0  0  0  0  0  0  0  0  0 11
12: 0  0  6  0  0  0  0  0  0  0  0 12
13: 0  0  0  0  0  0  0  0  0  0  0  0 13
14: 0  0  0  0  0  0  0  0  0  0  0  0  0 14
15: 0  0  0  0  0  0  0  0  0  0  0  0  0  0 15
16: 4  0  0  8  0  0  0  0 12  0  0  0  0  0  0 16
...
a(8,2) = sqrt(16) = 4, a(8,8) = sqrt(64) = 8, h^2 == 0 (mod 8) has A000188(8) = 2 solutions from 1 <= h <= 8, namely h = 4 and h = 8.
		

Links

• Wolfdieter Lang, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A008833, A000188.

Formula

a(n,m) = sqrt(n*m) > 0 if this is an integer and otherwise 0, for 1 <= m <= n. Due to commutativity this restriction is sufficient.