A132188 Number of 3-term geometric progressions with no term exceeding n.

1, 2, 3, 6, 7, 8, 9, 12, 17, 18, 19, 22, 23, 24, 25, 32, 33, 38, 39, 42, 43, 44, 45, 48, 57, 58, 63, 66, 67, 68, 69, 76, 77, 78, 79, 90, 91, 92, 93, 96, 97, 98, 99, 102, 107, 108, 109, 116, 129, 138, 139, 142, 143, 148, 149, 152, 153, 154, 155, 158

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

Cf. A057918, A064368, A120486, A132189, A132345, A211422, A338894.

Cf. also A000982, A362931.

Sequence in context: A275884 A003605 A344128 * A326027 A255527 A316156

Adjacent sequences: A132185 A132186 A132187 * A132189 A132190 A132191

Keyword

nonn

Author

Gerry Myerson, Nov 21 2007

Status

approved