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

1,3

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

Links

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

Formula

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

Example

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.

Crossrefs

Cf. A008833, A000188.

Sequence in context: A260110 A261115 A216229 * A259827 A143161 A225853

Adjacent sequences: A224774 A224775 A224776 * A224778 A224779 A224780

Keyword

nonn,tabl

Author

Wolfdieter Lang, Apr 25 2013

Status

approved