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%I #88 Sep 29 2023 05:30:08
%S 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,
%T 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,
%U 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
%N a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, d|n} a(d).
%C 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}.
%C 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
%C 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
%C 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.
%C From _Gus Wiseman_, Aug 20 2020: (Start)
%C 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:
%C 1 2 4 6 8 12
%C 2/1 4/1 6/1 8/1 12/1
%C 4/2 6/2 8/2 12/2
%C 4/2/1 6/3 8/4 12/3
%C 6/2/1 8/2/1 12/4
%C 6/3/1 8/4/1 12/6
%C 8/4/2 12/2/1
%C 8/4/2/1 12/3/1
%C 12/4/1
%C 12/4/2
%C 12/6/1
%C 12/6/2
%C 12/6/3
%C 12/4/2/1
%C 12/6/2/1
%C 12/6/3/1
%C (End)
%D 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.
%D Juhani Karhumaki, Yury Lifshits and Wojciech Rytter, Tiling Periodicity, in Combinatorial Pattern Matching, Lecture Notes in Computer Science, Volume 4580/2007, Springer-Verlag.
%H Reinhard Zumkeller, <a href="/A067824/b067824.txt">Table = of n, a(n) for n = 1..10000</a>
%H Olivier Bodini and Eric Rivals, <a href="https://web.archive.org/web/20170810212217/http://www.lirmm.fr/~rivals/PUBLI/FILES/OB-ER-CPM06.pdf">Tiling an Interval of the Discrete Line</a>
%H Thomas Fink, <a href="https://arxiv.org/abs/1912.07979">Recursively divisible numbers</a>, arXiv:1912.07979 [math.NT], 2019. See Table 1 p. 8.
%H T. M. A. Fink, <a href="https://arxiv.org/abs/2307.09140">Properties of the recursive divisor function and the number of ordered factorizations</a>, arXiv:2307.09140 [math.NT], 2023.
%H Michael Greene and Robin Michaels, <a href="https://www.archim.org.uk/eureka/archive/Eureka-54.pdf">One Dimensional Tilings</a>, Eureka (Cambridge) 54 (1996), 4-13.
%H G. Hajos, <a href="https://doi.org/10.1007/bf02021311">Sur le problème de factorisation des groupes cycliques</a>, Acta Math. Acad. Sci. Hung., 1:189-195, 1950.
%H J. Karhumaki and Y. Lifshits, <a href="http://logic.pdmi.ras.ru/~yura/en/tiling.pdf">Tiling periodicity</a>.
%H M. Krasner and B. Ranulac, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k31562/f397.item">Sur une propriété des polynomes de la division du cercle</a>, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
%H Eric H. Rivals, <a href="http://www.lirmm.fr/~rivals/RESEARCH/TILING/">Tiling</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F a(n) = 2*A074206(n), n>1. - _Vladeta Jovovic_, Jul 03 2005
%F a(p^k) = 2^k for primes p. - _Reinhard Zumkeller_, Sep 03 2006
%F a(n) = Sum_{d|n} A002033(d - 1). - _Gus Wiseman_, Jul 13 2018
%F Dirichlet g.f.: zeta(s) / (2 - zeta(s)). - _Álvar Ibeas_, Dec 30 2018
%F G.f. A(x) satisfies: A(x) = x/(1 - x) + Sum_{k>=2} A(x^k). - _Ilya Gutkovskiy_, May 18 2019
%e a(12) = 1 + a(6) + a(4) + a(3) + a(2) + a(1)
%e = 1+(1+a(3)+a(2)+a(1))+(1+a(2)+a(1))+(1+a(1))+(1+a(1))+(1)
%e = 1+(1+(1+a(1))+(1+a(1))+1)+(1+(1+a(1))+1)+(1+1)+(1+1)+(1)
%e = 1+(1+(1+1)+(1+1)+1)+(1+(1+1)+1)+(1+1)+(1+1)+(1)
%e = 1 + 6 + 4 + 2 + 2 + 1 = 16.
%p a:= proc(n) option remember;
%p 1+add(a(d), d=numtheory[divisors](n) minus {n})
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Apr 17 2021
%t 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 *)
%o (Haskell)
%o a067824 n = 1 + sum (map a067824 [d | d <- [1..n-1], mod n d == 0])
%o -- _Reinhard Zumkeller_, Oct 13 2011
%o (PARI) A=vector(100);A[1]=1; for(n=2,#A,A[n]=1+sumdiv(n,d,A[d])); A \\ _Charles R Greathouse IV_, Nov 20 2012
%Y Cf. A000005, A001678, A003238, A107067, A107748, A167865, A316782.
%Y Cf. A122408 (fixed points).
%Y A001055 counts factorizations.
%Y A008480 counts maximal chains of divisors starting with n.
%Y A074206 counts chains of divisors from n to 1.
%Y A253249 counts nonempty chains of divisors.
%Y A337070 counts chains of divisors starting with A006939(n).
%Y A337071 counts chains of divisors starting with n!.
%Y A337256 counts chains of divisors.
%Y Cf. A001221, A001222, A002033, A124010, A337074, A337105.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Feb 08 2002
%E Entry revised by _N. J. A. Sloane_, Aug 27 2006
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