

A067824


a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, dn} a(d).


23



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

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^n1 and such that (x^n1)/ {(x1)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^n1 and such that (x^n1)/ {(x1)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 enriched seriesreduced planted achiral (or generalized Bethe) trees with positive integer leaves summing to n, where a rooted tree is seriesreduced 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


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 117128. Springer Verlag, 2006.
G. Hajos. Sur le probleme de factorisation des groupes cycliques. Acta Math. Acad. Sci. Hung., 1:189195, 1950.
Juhani Karhumaki, Yury Lifshits and Wojciech Rytter, Tiling Periodicity, in Combinatorial Pattern Matching, Lecture Notes in Computer Science, Volume 4580/2007, SpringerVerlag.
M. Krasner and B. Ranulak. Sur une propriété des polynômes de la division du cercle. Comptes Rendus Académie des Sciences Paris, 240:397399, 1937.


LINKS

R. 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.
J. Karhumaki and Y. Lifshits, Tiling periodicity.
Eric H. Rivals, Tiling
Index entries for sequences computed from exponents in factorization of n


FORMULA

a(n) = 2*A074206(n), n>1.  Vladeta Jovovic, Jul 03 2005
a(n) = Sum_{dn} 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.


MATHEMATICA

a[1]=1; a[n_] := a[n] = 1+Sum[If[Mod[n, d]==0, a[d], 0], {d, 1, n1}]; Array[a, 100] (* JeanFrançois Alcover, Apr 28 2011 *)


PROG

(Haskell)
a067824 n = 1 + sum (map a067824 [d  d < [1..n1], mod n d == 0])
 Reinhard Zumkeller, Oct 13 2011
(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


CROSSREFS

Cf. A000005, A001678, A003238, A107067, A107748, A167865, A316782.
Sequence in context: A071364 A278237 A328707 * A107067 A320389 A046801
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



