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A060831
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a(n) = Sum_{k=1..n} (number of odd divisors of k) (cf. A001227).
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35
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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
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OFFSET
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0,3
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COMMENTS
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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
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
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LINKS
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FORMULA
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a(n) = n + floor(n/3) + floor(n/5) + floor(n/7) + floor(n/9) + ...
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
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EXAMPLE
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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.
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)
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)
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MAPLE
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add(numtheory[tau](n-i+1), i=1..ceil(n/2)) ;
end proc:
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MATHEMATICA
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f[n_] := Sum[ -(-1^k)Floor[n/(2k - 1)], {k, n}]; Table[ f[n], {n, 0, 65}] (* Robert G. Wilson v, Jun 16 2006 *)
Accumulate[Table[Count[Divisors[n], _?OddQ], {n, 0, 70}]] (* Harvey P. Dale, Nov 26 2023 *)
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PROG
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(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)
(Python)
from math import isqrt
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
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CROSSREFS
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Zero together with the partial sums of A001227.
Cf. A000005, A001620, A006218, A168508, A235791, A236104, A237048, A237590, A237593, A245092, A279387, A286001.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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