Search: id:A002033
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%I A002033 M0131 N0053
%S A002033 1,1,1,2,1,3,1,4,2,3,1,8,1,3,3,8,1,8,1,8,3,3,1,20,2,3,4,8,1,13,1,16,3,
               3,
%T A002033 3,26,1,3,3,20,1,13,1,8,8,3,1,48,2,8,3,8,1,20,3,20,3,3,1,44,1,3,8,32,3,
%U A002033 13,1,8,3,13,1,76,1,3,8,8,3,13,1,48,8,3,1,44,3,3,3,20,1,44,3,8,3,3,3,112
%N A002033 Number of perfect partitions of n.
%C A002033 A perfect partition of n is one which contains just one partition of 
               every number less than n when repeated parts are regarded as indistinguishable. 
               Thus 1^n is a perfect partition for every n; and for n = 7, 4 1^3, 
               4 2 1, 2^3 1 and 1^7 are all perfect partitions. [Riordan]
%C A002033 Also number of ordered factorizations of n+1, see A074206.
%C A002033 Also number of gozinta chains from 1 to n (see A034776) [ David W. Wilson 
               ]
%C A002033 a(n) is the permanent of the n X n matrix with (i,j) entry = 1 if j|i+1 
               and = 0 otherwise. For n=3 the matrix is {{1, 1, 0}, {1, 0, 1}, {1, 
               1, 0}} with permanent = 2. - David Callan (callan(AT)stat.wisc.edu), 
               Oct 19 2005
%C A002033 Appears to be the number of permutations that contribute to the determinant 
               that gives the moebius function. Verified up to a(9). [From Mats 
               O. Granvik (mgranvik(AT)abo.fi), Sep 13 2008]
%C A002033 Dirichlet inverse of A153881. [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Jan 03 2009]
%D A002033 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126, see #27.
%D A002033 R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 141.
%D A002033 HoKyu Lee, Double perfect partitions, Discrete Math., 306 (2006), 519-525.
%D A002033 P. A. MacMahon, The theory of perfect partitions and the compositions 
               of multipartite numbers, Messenger Math., 20 (1891), 103-119.
%D A002033 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 
               123-124.
%D A002033 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002033 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002033 T. D. Noe, <a href="b002033.txt">Table of n, a(n) for n = 0..9999</a>
%H A002033 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">Sequences realized as Parker vectors ...</
               a>, J. Integer Seqs., Vol. 6, 2003.
%H A002033 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PerfectPartition.html">Link to a section of The World of Mathematics.</
               a>
%H A002033 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DirichletSeriesGeneratingFunction.html">Dirichlet Series Generating 
               Function</a>
%H A002033 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A002033 a(n) = sum of all a(i) such that i divides n and i < n (Clark Kimberling).
%F A002033 a(p^k)=2^(k-1).
%F A002033 a(n) = A067824(n)/2 for n>1; a(A122408(n)) = A122408(n)/2. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 03 2006
%F A002033 a(n-1) = sum of all a(i-1) such that i divides n and i < n. a(p^k-1)=2^(k-1). 
               a(n-1) = A067824(n)/2 for n > 1; a(A122408(n)-1) = A122408(n)/2. 
               - David Wasserman (dwasserm(AT)earthlink.net), Nov 14 2006
%p A002033 a := array(1..150): for k from 1 to 150 do a[k] := 0 od: a[1] := 1: for 
               j from 2 to 150 do for m from 1 to j-1 do if j mod m = 0 then a[j] 
               := a[j]+a[m] fi: od: od: for k from 1 to 150 do printf(`%d,`,a[k]) 
               od: # from James A. Sellers Dec 07 2000
%o A002033 (PARI) A002033(n) = if(n==1,1,sumdiv(n,i,if(i==n,0,A002033(i)))) [From 
               Michael Porter (michael_b_porter(AT)yahoo.com), Nov 01 2009]
%Y A002033 Apart from initial term, same as A074206. Cf. A001055, A050324. a(A002110)=A000670.
%Y A002033 Cf. A000123, A100529, A117621.
%Y A002033 Sequence in context: A079616 A097283 A118314 this_sequence A074206 A108466 
               A087145
%Y A002033 Adjacent sequences: A002030 A002031 A002032 this_sequence A002034 A002035 
               A002036
%K A002033 nonn,core,easy,nice
%O A002033 0,4
%A A002033 N. J. A. Sloane (njas(AT)research.att.com).

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