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%I #114 Nov 26 2023 13:34:55
%S 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,
%T 52,54,58,60,62,66,68,69,73,75,79,82,84,86,90,92,94,98,100,102,108,
%U 110,112,114,117,120,124,126,128,132,136,138,142,144,146,150,152,154,160
%N a(n) = Sum_{k=1..n} (number of odd divisors of k) (cf. A001227).
%C The old definition was "Number of sums less than or equal to n of sequences of consecutive positive integers (including sequences of length 1)."
%C 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
%C Starting with 1 = row sums of triangle A168508. - _Gary W. Adamson_, Nov 27 2009
%C 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
%C 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
%C 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
%C Zero together with the row sums of A235791. - _Omar E. Pol_, Dec 18 2020
%H Harry J. Smith, <a href="/A060831/b060831.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{i=1..n} A001227(i).
%F a(n) = a(n-1) + A001227(n).
%F a(n) = n + floor(n/3) + floor(n/5) + floor(n/7) + floor(n/9) + ...
%F a(n) = A006218(n) - A006218(floor(n/2)).
%F a(n) = Sum_{i=1..ceiling(n/2)} A000005(n-i+1). - _Wesley Ivan Hurt_, Sep 30 2013
%F a(n) = Sum_{i=floor((n+2)/2)..n} A000005(i). - _N. J. A. Sloane_, Dec 06 2020, modified by _Xiaohan Zhang_, Nov 07 2022
%F G.f.: (1/(1 - x))*Sum_{k>=1} x^k/(1 - x^(2*k)). - _Ilya Gutkovskiy_, Dec 23 2016
%F a(n) ~ n*(log(2*n) + 2*gamma - 1) / 2, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 30 2019
%e 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
%e 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.
%e From _Omar E. Pol_, Dec 02 2020: (Start)
%e Illustration of initial terms:
%e Diagram
%e n a(n)
%e 0 0 _|
%e 1 1 _|1|
%e 2 2 _|1 _|
%e 3 4 _|1 |1|
%e 4 5 _|1 _| |
%e 5 7 _|1 |1 _|
%e 6 9 _|1 _| |1|
%e 7 11 _|1 |1 | |
%e 8 12 _|1 _| _| |
%e 9 15 _|1 |1 |1 _|
%e 10 17 _|1 _| | |1|
%e 11 19 _|1 |1 _| | |
%e 12 21 |1 | |1 | |
%e ...
%e 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)
%e From _Omar E. Pol_, Dec 19 2020: (Start)
%e 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):
%e . _ _ _ _
%e . _ _ _ |_ _ _ |_
%e . _ _ _ |_ _ _| |_ _ _| |_|_
%e . _ _ |_ _ |_ |_ _ |_ _ |_ _ |_ _ |
%e . _ _ |_ _|_ |_ _|_ | |_ _|_ | | |_ _|_ | | |
%e . _ |_ | |_ | | |_ | | | |_ | | | | |_ | | | | |
%e . |_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
%e .
%e . 0 1 2 4 5 7 9
%e (End)
%p A060831 := proc(n)
%p add(numtheory[tau](n-i+1),i=1..ceil(n/2)) ;
%p end proc:
%p seq(A060831(n),n=0..100) ; # _Wesley Ivan Hurt_, Oct 02 2013
%t f[n_] := Sum[ -(-1^k)Floor[n/(2k - 1)], {k, n}]; Table[ f[n], {n, 0, 65}] (* _Robert G. Wilson v_, Jun 16 2006 *)
%t Accumulate[Table[Count[Divisors[n],_?OddQ],{n,0,70}]] (* _Harvey P. Dale_, Nov 26 2023 *)
%o (PARI) a(n)=local(c);c=0;for(i=1,n,c+=sumdiv(i,X,X%2));c
%o (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
%o (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
%o (Python)
%o def A060831(n): return n+sum(n//i for i in range(3,n+1,2)) # _Chai Wah Wu_, Jul 16 2022
%o (Python)
%o from math import isqrt
%o def A060831(n): return ((t:=isqrt(m:=n>>1))+(s:=isqrt(n)))*(t-s)+(sum(n//k for k in range(1,s+1))-sum(m//k for k in range(1,t+1))<<1) # _Chai Wah Wu_, Oct 23 2023
%Y Zero together with the partial sums of A001227.
%Y Cf. A000005, A001620, A006218, A168508, A235791, A236104, A237048, A237590, A237593, A245092, A279387, A286001.
%K nonn
%O 0,3
%A _Henry Bottomley_, May 01 2001
%E Definition simplified by _N. J. A. Sloane_, Dec 05 2020
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