A135013 Partial sums of A000265.

1, 2, 5, 6, 11, 14, 21, 22, 31, 36, 47, 50, 63, 70, 85, 86, 103, 112, 131, 136, 157, 168, 191, 194, 219, 232, 259, 266, 295, 310, 341, 342, 375, 392, 427, 436, 473, 492, 531, 536, 577, 598, 641, 652, 697, 720, 767, 770, 819, 844, 895, 908, 961, 988, 1043, 1050, 1107, 1136

Offset

1,2

Comments

a(n) is also the number of elements in the set {(x,y): 1<=x,y<=n, the fraction x/y reduces to a fraction of the form (odd#)/(odd#)}. - Adam McDougall (mcdougall.adam(AT)gmail.com), Feb 20 2009

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

A. Erickson and F. Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.

R. A. MacLeod, On the Largest Odd Divisor of n, The American Mathematical Monthly, Vol. 75, No. 6 (Jun. - Jul., 1968), pp. 647-648.

Formula

a(n) = Sum_{k>=1} (round(n/2^k))^2. - Alejandro Erickson, Apr 13 2012

a(n) = n^2/3 + O(n) (see MacLeod link). - Michel Marcus, Dec 05 2013

a(j*2^k) = a(j) + (4^k-1)*j^2/3 for any j >= 1, k >= 0. - Jinyuan Wang, Mar 23 2019

Mathematica

Accumulate[Table[Times@@(#[[1]]^#[[2]]&/@Select[FactorInteger[i], #[[1]] != 2&]), {i, 90}]] (* Harvey P. Dale, Jun 25 2013 *)

Prog

(HP 50G Calculator) IDIV2 returns quotient & remainder to stack.
<< 0 SWAP
WHILE DUP 0 >
REPEAT 2 IDIV2 OVER + SQ ROT + SWAP
END DROP >>
# Gerald Hillier, May 02 2009, May 18 2009, Aug 01 2009
(PARI) a(n)=sum(k=1, log(n)\log(2)+1, round(n/2^k)^2) \\ Charles R Greathouse IV, Oct 06 2013

Crossrefs

Cf. A000918, A000265, A135014.

Sequence in context: A340001 A341446 A015613 * A239447 A336527 A293398

Adjacent sequences: A135010 A135011 A135012 * A135014 A135015 A135016

Keyword

nonn

Author

N. J. A. Sloane, Feb 10 2008

Status

approved