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

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, 19, 19, 19, 22, 22, 22, 22, 27, 27, 27, 27, 28, 28, 28, 28, 29, 31, 31, 31, 34, 40, 44, 44, 45, 45, 47, 47, 48, 48, 48, 48, 49, 49, 49, 51, 58, 58, 58, 58, 59, 59, 59, 59, 64, 64, 64, 68

Offset

1,8

Comments

A078147 gives run lengths in this sequence, apart from initial run of zeros. - Reinhard Zumkeller, Apr 22 2012

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

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.

a(n) = A132188(n) - A120486(n). - Reinhard Zumkeller, Apr 22 2012

Example

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

Maple

sum(numtheory[phi](p)*trunc(n/p^2), p=2..trunc(sqrt(n)));

Prog

(Haskell)
a132345 n = sum $ zipWith (*)
   (tail a000010_list) (map ((div n) . (^ 2)) [2..a000196 n])
-- Reinhard Zumkeller, Apr 22 2012

Crossrefs

Cf. A000010, A000196, A057918 (first differences).

Sequence in context: A097154 A108421 A104058 * A178976 A352426 A130766

Adjacent sequences: A132342 A132343 A132344 * A132346 A132347 A132348

Keyword

easy,nonn

Author

David Angell (angell(AT)maths.unsw.edu.au), Nov 07 2007

Status

approved