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

Author

N. J. A. Sloane, Feb 10 2008

Keywords

• nonn

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.

Crossrefs

Cf. A000918, A000265, A135014.

Programs

Mathematica

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

PARI

a(n)=sum(k=1,log(n)\log(2)+1,round(n/2^k)^2) \\ Charles R Greathouse IV, Oct 06 2013

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