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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.

%I A224777 #18 Jan 11 2025 03:15:52
%S A224777 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,
%T A224777 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,
%U A224777 0,6,0,0,0,0,0,0,0,0,12,0,0
%N A224777 Triangle with integer geometric mean sqrt(n*m) for 1 <= m <= n, and 0 if sqrt(n*m) is not integer.
%C A224777 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.
%C A224777 The number of nonzero entries in row n is A000188(n).
%C A224777 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
%H A224777 Wolfdieter Lang, <a href="/A224777/b224777.txt">Rows n = 1..100 of triangle, flattened</a>
%F A224777 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.
%e A224777 The triangle begins:
%e A224777 n\m 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
%e A224777 1:  1
%e A224777 2:  0  2
%e A224777 3:  0  0  3
%e A224777 4:  2  0  0  4
%e A224777 5:  0  0  0  0  5
%e A224777 6:  0  0  0  0  0  6
%e A224777 7:  0  0  0  0  0  0  7
%e A224777 8:  0  4  0  0  0  0  0  8
%e A224777 9:  3  0  0  6  0  0  0  0  9
%e A224777 10: 0  0  0  0  0  0  0  0  0 10
%e A224777 11: 0  0  0  0  0  0  0  0  0  0 11
%e A224777 12: 0  0  6  0  0  0  0  0  0  0  0 12
%e A224777 13: 0  0  0  0  0  0  0  0  0  0  0  0 13
%e A224777 14: 0  0  0  0  0  0  0  0  0  0  0  0  0 14
%e A224777 15: 0  0  0  0  0  0  0  0  0  0  0  0  0  0 15
%e A224777 16: 4  0  0  8  0  0  0  0 12  0  0  0  0  0  0 16
%e A224777 ...
%e A224777 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.
%Y A224777 Cf. A008833, A000188.
%K A224777 nonn,tabl
%O A224777 1,3
%A A224777 _Wolfdieter Lang_, Apr 25 2013