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

%I A132188 #36 Aug 29 2023 11:00:11
%S A132188 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,
%T A132188 58,63,66,67,68,69,76,77,78,79,90,91,92,93,96,97,98,99,102,107,108,
%U A132188 109,116,129,138,139,142,143,148,149,152,153,154,155,158
%N A132188 Number of 3-term geometric progressions with no term exceeding n.
%C A132188 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
%C A132188 Also the number of 2 X 2 symmetric singular matrices with entries from {1, ..., n} - cf. A064368.
%C A132188 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
%H A132188 Alois P. Heinz, <a href="/A132188/b132188.txt">Table of n, a(n) for n = 1..10000</a>
%H A132188 Gerry Myerson, <a href="https://austms.org.au/wp-content/uploads/Gazette/2008/Jul08/Gazette35(3)Web.pdf">Trifectas in Geometric Progression</a>, Australian Mathematical Society Gazette 35 (3) (2008) pp. 189--194 (pages 47--52 in PDF).
%F A132188 a(n) = Sum [sqrt(n/k)]^2, where the sum is over all squarefree k not exceeding n.
%F A132188 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).
%F A132188 a(n) = Sum_{1 <=x,y <=n} A010052(x*y). - _Clark Kimberling_, Apr 14 2012
%F A132188 a(n) = n+2*Sum_{1<=x<y<=n} A010052(x*y). - _Chai Wah Wu_, Aug 28 2023
%e A132188 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
%p A132188 a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
%p A132188       1+2*add(`if`(issqr(i*n), 1, 0), i=1..n-1))
%p A132188     end:
%p A132188 seq(a(n), n=1..60);  # _Alois P. Heinz_, Aug 28 2023
%t A132188 t[n_] := t[n] = Flatten[Table[w^2 - x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
%t A132188 c[n_] := Count[t[n], 0]
%t A132188 t = Table[c[n], {n, 0, 80}]  (* _Clark Kimberling_, Apr 14 2012 *)
%o A132188 (Haskell)
%o A132188 a132188 0 = 0
%o A132188 a132188 n = a132345 n + (a120486 $ fromInteger n)
%o A132188 -- _Reinhard Zumkeller_, Apr 21 2012
%o A132188 (Python)
%o A132188 from sympy.ntheory.primetest import is_square
%o A132188 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
%Y A132188 Cf. A057918, A064368, A120486, A132189, A132345, A211422, A338894.
%Y A132188 Cf. also A000982, A362931.
%K A132188 nonn
%O A132188 1,2
%A A132188 _Gerry Myerson_, Nov 21 2007