The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A067824 a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, d|n} a(d). 88
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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}. a(p^k) = 2^k for primes p; a(n) = n iff n = A122408(k) for some k. - Reinhard Zumkeller, Sep 03 2006 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 Ranulak, 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. From Gus Wiseman, Aug 20 2020: (Start) 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) REFERENCES 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. LINKS Reinhard Zumkeller, Table = of n, a(n) for n = 1..10000 Olivier Bodini and Eric Rivals, Tiling an Interval of the Discrete Line Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. See Table 1 p. 8. G. Hajos, Sur le problème de factorisation des groupes cycliques Acta Math. Acad. Sci. Hung., 1:189-195, 1950. J. Karhumaki and Y. Lifshits, Tiling periodicity. M. Krasner and B. Ranulak, Sur une propriété des polynômes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937. Eric H. Rivals, Tiling FORMULA a(n) = 2*A074206(n), n>1. - Vladeta Jovovic, Jul 03 2005 a(n) = Sum_{d|n} A002033(d - 1). - Gus Wiseman, Jul 13 2018 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 EXAMPLE 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. MAPLE a:= proc(n) option remember;       1+add(a(d), d=numtheory[divisors](n) minus {n})     end: seq(a(n), n=1..100);  # Alois P. Heinz, Apr 17 2021 MATHEMATICA a=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 *) PROG (Haskell) a067824 n = 1 + sum (map a067824 [d | d <- [1..n-1], mod n d == 0]) -- Reinhard Zumkeller, Oct 13 2011 (PARI) A=vector(100); A=1; for(n=2, #A, A[n]=1+sumdiv(n, d, A[d])); A \\ Charles R Greathouse IV, Nov 20 2012 CROSSREFS Cf. A000005, A001678, A003238, A107067, A107748, A167865, A316782. A001055 counts factorizations. A008480 counts maximal chains of divisors starting with n. A074206 counts chains of divisors from n to 1. A253249 counts nonempty chains of divisors. A337070 counts chains of divisors starting with A006939(n). A337071 counts chains of divisors starting with n!. A337256 counts chains of divisors. Cf. A001221, A001222, A002033, A124010, A337074, A337105. Sequence in context: A071364 A278237 A328707 * A347458 A107067 A331580 Adjacent sequences:  A067821 A067822 A067823 * A067825 A067826 A067827 KEYWORD nonn AUTHOR Reinhard Zumkeller, Feb 08 2002 EXTENSIONS Entry revised by N. J. A. Sloane, Aug 27 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 28 07:52 EST 2021. Contains 349401 sequences. (Running on oeis4.)