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A224777
Triangle with integer geometric mean sqrt(n*m) for 1 <= m <= n, and 0 if sqrt(n*m) is not integer.
1
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
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
Sequence in context: A260110 A261115 A216229 * A259827 A143161 A225853
KEYWORD
nonn,tabl
AUTHOR
Wolfdieter Lang, Apr 25 2013
STATUS
approved