OFFSET
1,2
COMMENTS
a(n) = number of pairs (i,j) in [1..n] X [1..n] with integral geometric mean sqrt(i*j). Cf. A000982, A362931. - N. J. A. Sloane, Aug 28 2023
Also the number of 2 X 2 symmetric singular matrices with entries from {1, ..., n} - cf. A064368.
Rephrased: Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=x*y. See A211422. - Clark Kimberling, Apr 14 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette 35 (3) (2008) pp. 189--194 (pages 47--52 in PDF).
FORMULA
a(n) = Sum [sqrt(n/k)]^2, where the sum is over all squarefree k not exceeding n.
If we call A120486, this sequence and A132189 F(n), P(n) and S(n), respectively, then P(n) = 2 F(n) - n = S(n) + n. The Finch-Sebah paper cited at A000188 proves that F(n) is asymptotic to (3 / pi^2) n log n. In the reference, we prove that F(n) = (3 / pi^2) n log n + O(n), from which it follows that P(n) = (6 / pi^2) n log n + O(n) and similarly for S(n).
a(n) = Sum_{1 <=x,y <=n} A010052(x*y). - Clark Kimberling, Apr 14 2012
a(n) = n+2*Sum_{1<=x<y<=n} A010052(x*y). - Chai Wah Wu, Aug 28 2023
EXAMPLE
a(4) counts these six (w,x,y) - triples: (1,1,1), (2,1,4), (2,4,1), (2,2,2), (3,3,3), (4,4,4). - Clark Kimberling, Apr 14 2012
MAPLE
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
1+2*add(`if`(issqr(i*n), 1, 0), i=1..n-1))
end:
seq(a(n), n=1..60); # Alois P. Heinz, Aug 28 2023
MATHEMATICA
t[n_] := t[n] = Flatten[Table[w^2 - x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}] (* Clark Kimberling, Apr 14 2012 *)
PROG
(Haskell)
a132188 0 = 0
a132188 n = a132345 n + (a120486 $ fromInteger n)
-- Reinhard Zumkeller, Apr 21 2012
(Python)
from sympy.ntheory.primetest import is_square
def A132188(n): return n+(sum(1 for x in range(1, n+1) for y in range(1, x) if is_square(x*y))<<1) # Chai Wah Wu, Aug 28 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gerry Myerson, Nov 21 2007
STATUS
approved