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

Author

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

Keywords

• easy
• nonn

Comments

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

Examples

			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
		

Links

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

Crossrefs

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

Cf. A120486, A132188, A132189.

Programs

Haskell

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

Maple

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

Formula

a(n)=sum_{1

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

a(n) = (A132188(n) - n)/2 = A120486(n) - n. - David Radcliffe, Jun 08 2025

a(n) = A132189(n)/2. - Hugo Pfoertner, Jun 08 2025

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