cp's OEIS Frontend

A132345 Number of increasing three-term geometric sequences from the integers {1,2,...,n}.

%I A132345 #19 Jun 08 2025 15:10:20
%S A132345 0,0,0,1,1,1,1,2,4,4,4,5,5,5,5,8,8,10,10,11,11,11,11,12,16,16,18,19,
%T A132345 19,19,19,22,22,22,22,27,27,27,27,28,28,28,28,29,31,31,31,34,40,44,44,
%U A132345 45,45,47,47,48,48,48,48,49,49,49,51,58,58,58,58,59,59,59,59,64,64,64,68
%N A132345 Number of increasing three-term geometric sequences from the integers {1,2,...,n}.
%C A132345 A078147 gives run lengths in this sequence, apart from initial run of zeros. - _Reinhard Zumkeller_, Apr 22 2012
%H A132345 Reinhard Zumkeller, <a href="/A132345/b132345.txt">Table of n, a(n) for n = 1..10000</a>
%F A132345 a(n)=sum_{1<p^2<=n} phi(p)trunc(n/p^2) where phi is Euler's phi function and trunc is the greatest integer function.
%F A132345 a(n) = A132188(n) - A120486(n). - _Reinhard Zumkeller_, Apr 22 2012
%F A132345 a(n) = (A132188(n) - n)/2 = A120486(n) - n. - _David Radcliffe_, Jun 08 2025
%F A132345 a(n) = A132189(n)/2. - _Hugo Pfoertner_, Jun 08 2025
%e A132345 a(24)=12 as the sequences counted are 1,2,4; 2,4,8; 3,6,12; 4,8,16; 5,10,20; 6,12,24; 1,3,9; 2,6,18; 4,6,9; 8,12,18; 1,4,16; 9,12,16
%p A132345 sum(numtheory[phi](p)*trunc(n/p^2),p=2..trunc(sqrt(n)));
%o A132345 (Haskell)
%o A132345 a132345 n = sum $ zipWith (*)
%o A132345    (tail a000010_list) (map ((div n) . (^ 2)) [2..a000196 n])
%o A132345 -- _Reinhard Zumkeller_, Apr 22 2012
%Y A132345 Cf. A000010, A000196, A057918 (first differences).
%Y A132345 Cf. A120486, A132188, A132189.
%K A132345 easy,nonn
%O A132345 1,8
%A A132345 David Angell (angell(AT)maths.unsw.edu.au), Nov 07 2007