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A132188 Number of 3-term geometric progressions with no term exceeding n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Table of n, a(n) for n=1..60.

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( A010052(x*y): 1 <=x,y <=n ). - Clark Kimberling, Apr 14 2012

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

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

CROSSREFS

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

Sequence in context: A102806 A275884 A003605 * A326027 A255527 A316156

Adjacent sequences:  A132185 A132186 A132187 * A132189 A132190 A132191

KEYWORD

nonn

AUTHOR

Gerry Myerson, Nov 21 2007

STATUS

approved

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Last modified March 3 13:38 EST 2021. Contains 341762 sequences. (Running on oeis4.)