Search: id:A006218
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%I A006218 M2432
%S A006218 0,1,3,5,8,10,14,16,20,23,27,29,35,37,41,45,50,52,58,60,66,70,74,76,84,
%T A006218 87,91,95,101,103,111,113,119,123,127,131,140,142,146,150,158,160,168,
%U A006218 170,176,182,186,188,198,201,207,211,217,219,227,231,239,243,247,249
%N A006218 Sum_{k=1..n} floor(n/k); also sum_{k=1..n} d(k), where d = number of 
               divisors (A000005); also number of solutions to xy = z with 1 <= 
               x,y,z <= n.
%C A006218 The "Dirichlet divisor problem" is to find a precise asymptotic estimate 
               for this sequence - see formula lines below.
%C A006218 Number of increasing arithmetical progressions where n+1 is the second 
               or later term. - Mambetov Timur, Takenov Nurdin, Haritonova Oksana 
               (timus(AT)post.kg; oksanka-61(AT)mail.ru), Jun 13 2002. E.g. a(3)=5 
               because there are 5 such arithmetical progressions: (1,2,3,4); (2,
               3,4); (1,4); (2,4); (3,4).
%C A006218 Binomial transform of A001659.
%C A006218 Area covered by overlapped partitions of n, i.e. sum of maximum values 
               of the k-th part of a partition of n into k parts - Jon Perry (perry(AT)globalnet.co.uk), 
               Sep 08 2005
%C A006218 Equals inverse Mobius transform of A116447 (i.e. A051731 * A116477, for 
               n>0). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2008]
%C A006218 Equals row sums of triangle A143724 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 29 2008]
%D A006218 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006218 K. Chandrasekharan, Introduction to Analytic Number Theory, Sringer-Verlag, 
               1968, Chap. VI.
%D A006218 K. Chandrasekharan, (1970): Arithmetical Functions; Chapter VIII, pp. 
               194-228. Springer-Verlag, Berlin.
%D A006218 P. G. L. Dirichlet, Werke, Vol. ii, pp. 49-66.
%D A006218 L. Hoehn and J. Ridenhour, Summations involving computer-related functions, 
               Math. Mag., 62 (1989), 191-196.
%D A006218 M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 
               1972, p. 7.
%D A006218 M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; 
               p. 239.
%D A006218 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number 
               Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 56.
%D A006218 Takenov, N. N. and Haritonova, O., Representation of positive integers 
               by a special set of digits and sequences, in Dolmatov, S. L. et al. 
               editors, Materials of Science, Practical seminar "Modern Mathematics."
%D A006218 R. Bellman and H. N. Shapiro, On a problem in additive number theory, 
               Annals Math., 49 (1948), 333-340. See Eq. 1.5. [From N. J. A. Sloane, 
               Mar 12 2009]
%H A006218 T. D. Noe, <a href="b006218.txt">Table of n, a(n) for n = 0..1000</a>
%F A006218 Sum_{k=1..n} d(k) = n * ( log(n) + 2*gamma - 1 ) + O(sqrt(n)), where 
               gamma is the Euler-Mascheroni number ~ 0.57721... (see A001620), 
               Dirichlet, 1849. Again, sum_{k=1..n} d(k) = n * ( log(n) + 2*gamma 
               - 1 ) + O(log(n)*n^(1/3)). The determination of the precise size 
               of the error term is an unsolved problem - see references.
%F A006218 The bounds from Chandrasekharan lead to the explicit bounds n log(n) 
               + (2 gamma - 1) n - 4 sqrt(n) - 1 <= a(n) <= n log(n) + (2 gamma 
               - 1) n + 4 sqrt(n) [From David Applegate (david(AT)research.att.com), 
               Oct 14 2008]
%F A006218 a(n)=2*sum(i=1, floor(sqrt(n)), floor(n/i))-floor(sqrt(n))^2 - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), May 12 2002
%F A006218 G.f.: 1/(1-x)*sum(k>=1, x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 23 2003
%F A006218 For n>0: A027750(a(n-1) + k) = k-divisor of n, = k<=A000005(n). - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), May 10 2006
%F A006218 a(n) = A161886(n) - n + 1 = A161886(n-1) - A049820(n) + 2 = A161886(n-1) 
               + A000005(n) - n + 2 = A006590(n) + A000005(n) - n = A006590(n+1) 
               - n - 1 = A006590(n) + A000005(n) - n for n >= 2. a(n) = a(n-1) + 
               A000005(n) for n >= 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Nov 14 2009]
%p A006218 with(numtheory): A006218 := n->add(sigma[0](i),i=1..n);
%p A006218 with(numtheory):a[1]:=1: for n from 2 to 59 do a[n]:=a[n-1]+tau(n) od: 
               seq(a[n], n=0..59);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 21 2009]
%t A006218 Table[Sum[DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]
%t A006218 FoldList[Plus, 0, Table[DivisorSigma[0, x], {x, 61}]]//Rest (much faster)
%o A006218 (PARI) a(n)=sum(k=1,n,n\k)
%Y A006218 Right edge of A056535. Cf. A000005, A001659, A052511, A143236.
%Y A006218 Row sums of triangle A003988.
%Y A006218 A061017 is an inverse.
%Y A006218 It appears that the partial sums give A078567. - N. J. A. Sloane (njas(AT)research.att.com), 
               Nov 24 2008
%Y A006218 Cf. A116477, A051731, A143724 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 07 2008]
%Y A006218 A161700. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 17 2009]
%Y A006218 Sequence in context: A027922 A051611 A005004 this_sequence A062839 A088940 
               A088937
%Y A006218 Adjacent sequences: A006215 A006216 A006217 this_sequence A006219 A006220 
               A006221
%K A006218 nonn,easy,nice
%O A006218 0,3
%A A006218 N. J. A. Sloane (njas(AT)research.att.com).
%E A006218 Comment about explicit bounds from Chandrasekharan [From David Applegate 
               (david(AT)research.att.com), Oct 14 2008]

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