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A120486 Partial sums of A000188.

%I A120486 #46 Oct 25 2024 10:13:13
%S A120486 1,2,3,5,6,7,8,10,13,14,15,17,18,19,20,24,25,28,29,31,32,33,34,36,41,
%T A120486 42,45,47,48,49,50,54,55,56,57,63,64,65,66,68,69,70,71,73,76,77,78,82,
%U A120486 89,94,95,97,98,101,102,104,105,106,107,109,110,111,114,122,123,124,125,127,128
%N A120486 Partial sums of A000188.
%C A120486 This sequence can also be described as the number of 3-term nondecreasing geometric progressions with no term exceeding n.
%C A120486 a(n) = A132188(n) - A132345(n). - _Reinhard Zumkeller_, Apr 21 2012
%H A120486 Reinhard Zumkeller, <a href="/A120486/b120486.txt">Table of n, a(n) for n = 1..10000</a>
%H A120486 Vaclav Kotesovec, <a href="/A120486/a120486.jpg">Graph - the asymptotic ratio</a>
%H A120486 Gerry Myerson, <a href="http://www.austms.org.au/Publ/Gazette/2008/Jul08/TechPaperMyerson.pdf">Trifectas in Geometric Progression</a>, Australian Mathematical Society Gazette, 35 (3) 2008, p 189-194.
%F A120486 a(n) = 3n log(n) / Pi^2 + O(n). - _Griffin N. Macris_, Jan 28 2017
%F A120486 a(n) ~ 3*n*((log(n) + (3*gamma - 1))/ Pi^2 - 12*(Zeta'(2)/Pi^4)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 30 2019
%F A120486 a(n) = Sum_{k=1..floor(sqrt(n))} phi(k)*floor(n/k^2), where phi is the Euler totient function A000010. - _Ridouane Oudra_, Aug 18 2019
%F A120486 G.f.: (1/(1 - x)) * Sum_{k>=1} phi(k) * x^(k^2) / (1 - x^(k^2)). - _Ilya Gutkovskiy_, Aug 26 2021
%F A120486 From _Ridouane Oudra_, Oct 05 2024: (Start)
%F A120486 a(n) = Sum_{i=1..n} Sum_{j=1..i} A010052(i*j).
%F A120486 a(n) = A132345(n) + n.
%F A120486 a(n) = (1/2)*A132189(n) + n.
%F A120486 a(n) = (1/2)*(A132188(n) + n). (End)
%p A120486 with(numtheory): seq(add(phi(k)*floor(n/k^2), k=1..floor(sqrt(n))), n=1..100); # _Ridouane Oudra_, Aug 18 2019
%o A120486 (Haskell)
%o A120486 a120486 n = a120486_list !! (n - 1)
%o A120486 a120486_list = scanl1 (+) a000188_list
%o A120486 -- _Reinhard Zumkeller_, Apr 22 2012
%Y A120486 Cf. A000188, A132188, A132189, A132345, A010052.
%K A120486 nonn
%O A120486 1,2
%A A120486 _Gerry Myerson_, Nov 21 2007