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 A060831 a(n) = Sum_{k=1..n} (number of odd divisors of k) (cf. A001227). 33
 0, 1, 2, 4, 5, 7, 9, 11, 12, 15, 17, 19, 21, 23, 25, 29, 30, 32, 35, 37, 39, 43, 45, 47, 49, 52, 54, 58, 60, 62, 66, 68, 69, 73, 75, 79, 82, 84, 86, 90, 92, 94, 98, 100, 102, 108, 110, 112, 114, 117, 120, 124, 126, 128, 132, 136, 138, 142, 144, 146, 150, 152, 154, 160 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The old definition was "Number of sums less than or equal to n of sequences of consecutive positive integers (including sequences of length 1)." In other words, a(n) is also the total number of partitions of all positive integers <= n into consecutive parts, n >= 1. - Omar E. Pol, Dec 03 2020 Starting with 1 = row sums of triangle A168508. - Gary W. Adamson, Nov 27 2009 The subsequence of primes in this sequence begins, through a(100): 2, 5, 7, 11, 17, 19, 23, 29, 37, 43, 47, 73, 79, 173, 181, 223, 227, 229, 233, 263. - Jonathan Vos Post, Feb 13 2010 Apart from the initial zero, a(n) is also the total number of subparts of the symmetric representations of sigma of all positive integers <= n. Hence a(n) is also the total number of subparts in the terraces of the stepped pyramid with n levels described in A245092. For more information see A279387 and A237593. - Omar E. Pol, Dec 17 2016 a(n) is also the total number of partitions of all positive integers <= n into an odd number of equal parts. - Omar E. Pol, May 14 2017 Zero together with the row sums of A235791. - Omar E. Pol, Dec 18 2020 LINKS Harry J. Smith, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{i=1..n} A001227(i). a(n) = a(n-1) + A001227(n). a(n) = n + floor(n/3) + floor(n/5) + floor(n/7) + floor(n/9) + ... a(n) = A006218(n) - A006218(floor(n/2)). a(n) = Sum_{i=1..ceiling(n/2)} A000005(n-i+1). - Wesley Ivan Hurt, Sep 30 2013 Or, equivalently, a(n) = Sum_{j=1..floor((n+2)/2)} d(j), where d(j) is the number of divisors function A000005. - N. J. A. Sloane, Dec 06 2020 G.f.: (1/(1 - x))*Sum_{k>=1} x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 23 2016 a(n) ~ n*(log(2*n) + 2*gamma - 1) / 2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 30 2019 EXAMPLE E.g., for a(7), we consider the odd divisors of 1,2,3,4,5,6,7, which gives 1,1,2,1,2,2,2 = 11. - Jon Perry, Mar 22 2004 Example illustrating the old definition: a(7) = 11 since 1, 2, 3, 4, 5, 6, 7, 1+2, 2+3, 3+4, 1+2+3 are all 7 or less. From Omar E. Pol, Dec 02 2020: (Start) Illustration of initial terms:                               Diagram    n   a(n)    0     0                          _|    1     1                        _|1|    2     2                      _|1 _|    3     4                    _|1  |1|    4     5                  _|1   _| |    5     7                _|1    |1 _|    6     9              _|1     _| |1|    7    11            _|1      |1  | |    8    12          _|1       _|  _| |    9    15        _|1        |1  |1 _|   10    17      _|1         _|   | |1|   11    19    _|1          |1   _| | |   12    21   |1            |   |1  | | ... a(n) is also the total number of horizontal line segments in the first n levels of the diagram. For n = 5 there are seven horizontal line segments, so a(5) = 7. Cf. A237048, A286001. (End) From Omar E. Pol, Dec 19 2020: (Start) a(n) is also the number of regions in the diagram of the symmetries of sigma after n stages, including the subparts, as shown below (Cf. A279387): .                                                         _ _ _ _ .                                           _ _ _        |_ _ _  |_ .                               _ _ _      |_ _ _|       |_ _ _| |_|_ .                     _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  | .             _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | | .       _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | | .      |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_| . .  0    1      2        4          5            7              9 (End) MAPLE A060831 := proc(n)     add(numtheory[tau](n-i+1), i=1..ceil(n/2)) ; end proc: seq(A060831(n), n=0..100) ; # Wesley Ivan Hurt, Oct 02 2013 MATHEMATICA f[n_] := Sum[ -(-1^k)Floor[n/(2k - 1)], {k, n}]; Table[ f[n], {n, 0, 65}] (* Robert G. Wilson v, Jun 16 2006 *) PROG (PARI) a(n)=local(c); c=0; for(i=1, n, c+=sumdiv(i, X, X%2)); c (PARI) for (n=0, 1000, s=n; d=3; while (n>=d, s+=n\d; d+=2); write("b060831.txt", n, " ", s); ) \\ Harry J. Smith, Jul 12 2009 (PARI) a(n)=my(n2=n\2); sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2-sum(k=1, sqrtint(n2), n2\k)*2+sqrtint(n2)^2 \\ Charles R Greathouse IV, Jun 18 2015 (Python) def A060831(n): return n+sum(n//i for i in range(3, n+1, 2)) # Chai Wah Wu, Jul 16 2022 CROSSREFS Zero together with the partial sums of A001227. Cf. A000005, A001620, A006218, A168508, A235791, A236104, A237048, A237590, A237593, A245092, A279387, A286001. Sequence in context: A325101 A301728 A005152 * A073727 A075692 A112235 Adjacent sequences:  A060828 A060829 A060830 * A060832 A060833 A060834 KEYWORD nonn AUTHOR Henry Bottomley, May 01 2001 EXTENSIONS Definition simplified by N. J. A. Sloane, Dec 05 2020 STATUS approved

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Last modified August 17 16:21 EDT 2022. Contains 356189 sequences. (Running on oeis4.)