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A067824
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a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, d|n} a(d).
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90
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1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 16, 2, 6, 6, 16, 2, 16, 2, 16, 6, 6, 2, 40, 4, 6, 8, 16, 2, 26, 2, 32, 6, 6, 6, 52, 2, 6, 6, 40, 2, 26, 2, 16, 16, 6, 2, 96, 4, 16, 6, 16, 2, 40, 6, 40, 6, 6, 2, 88, 2, 6, 16, 64, 6, 26, 2, 16, 6, 26, 2, 152, 2, 6, 16, 16, 6, 26, 2, 96, 16, 6, 2, 88, 6, 6, 6, 40, 2, 88, 6, 16, 6, 6, 6, 224, 2, 16, 16, 52
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OFFSET
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1,2
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COMMENTS
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By a result of Karhumaki and Lifshits, this is also the number of polynomials p(x) with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/ {(x-1)p(x)} has all coefficients in {0,1}.
The number of tiles of a discrete interval of length n (an interval of Z). - Eric H. Rivals (rivals(AT)lirmm.fr), Mar 13 2007
Bodini and Rivals proved this is the number of tiles of a discrete interval of length n and also is the number (A107067) of polynomials p(x) with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/ {(x-1)p(x)} has all coefficients in {0,1} (Bodini, Rivals, 2006). This structure of such tiles is also known as Krasner's factorization (Krasner and Ranulac, 1937). The proof also gives an algorithm to recognize if a set is a tile in optimal time and in this case, to compute the smallest interval it can tile (Bodini, Rivals, 2006). - Eric H. Rivals (rivals(AT)lirmm.fr), Mar 13 2007
Number of lone-child-avoiding rooted achiral (or generalized Bethe) trees with positive integer leaves summing to n, where a rooted tree is lone-child-avoiding if all terminal subtrees have at least two branches, and achiral if all branches directly under any given node are equal. For example, the a(6) = 6 trees are 6, (111111), (222), ((11)(11)(11)), (33), ((111)(111)). - Gus Wiseman, Jul 13 2018. Updated Aug 22 2020.
Also the number of strict chains of divisors starting with n. For example, the a(n) chains for n = 1, 2, 4, 6, 8, 12 are:
1 2 4 6 8 12
2/1 4/1 6/1 8/1 12/1
4/2 6/2 8/2 12/2
4/2/1 6/3 8/4 12/3
6/2/1 8/2/1 12/4
6/3/1 8/4/1 12/6
8/4/2 12/2/1
8/4/2/1 12/3/1
12/4/1
12/4/2
12/6/1
12/6/2
12/6/3
12/4/2/1
12/6/2/1
12/6/3/1
(End)
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REFERENCES
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Olivier Bodini and Eric Rivals. Tiling an Interval of the Discrete Line. In M. Lewenstein and G. Valiente, editors, Proc. of the 17th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 4009 of Lecture Notes in Computer Science, pages 117-128. Springer Verlag, 2006.
Juhani Karhumaki, Yury Lifshits and Wojciech Rytter, Tiling Periodicity, in Combinatorial Pattern Matching, Lecture Notes in Computer Science, Volume 4580/2007, Springer-Verlag.
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s) / (2 - zeta(s)). - Álvar Ibeas, Dec 30 2018
G.f. A(x) satisfies: A(x) = x/(1 - x) + Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, May 18 2019
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EXAMPLE
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a(12) = 1 + a(6) + a(4) + a(3) + a(2) + a(1)
= 1+(1+a(3)+a(2)+a(1))+(1+a(2)+a(1))+(1+a(1))+(1+a(1))+(1)
= 1+(1+(1+a(1))+(1+a(1))+1)+(1+(1+a(1))+1)+(1+1)+(1+1)+(1)
= 1+(1+(1+1)+(1+1)+1)+(1+(1+1)+1)+(1+1)+(1+1)+(1)
= 1 + 6 + 4 + 2 + 2 + 1 = 16.
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MAPLE
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a:= proc(n) option remember;
1+add(a(d), d=numtheory[divisors](n) minus {n})
end:
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MATHEMATICA
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a[1]=1; a[n_] := a[n] = 1+Sum[If[Mod[n, d]==0, a[d], 0], {d, 1, n-1}]; Array[a, 100] (* Jean-François Alcover, Apr 28 2011 *)
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PROG
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(Haskell)
a067824 n = 1 + sum (map a067824 [d | d <- [1..n-1], mod n d == 0])
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CROSSREFS
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A008480 counts maximal chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A253249 counts nonempty chains of divisors.
A337071 counts chains of divisors starting with n!.
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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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