0,3

The "Dirichlet divisor problem" is to find a precise asymptotic estimate for this sequence - see formula lines below.

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).

Binomial transform of A001659.

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

Equals inverse Mobius transform of A116447 (i.e. A051731 * A116477, for n>0). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2008]

Equals row sums of triangle A143724 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 29 2008]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

K. Chandrasekharan, Introduction to Analytic Number Theory, Sringer-Verlag, 1968, Chap. VI.

K. Chandrasekharan, (1970): Arithmetical Functions; Chapter VIII, pp. 194-228. Springer-Verlag, Berlin.

P. G. L. Dirichlet, Werke, Vol. ii, pp. 49-66.

L. Hoehn and J. Ridenhour, Summations involving computer-related functions, Math. Mag., 62 (1989), 191-196.

M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972, p. 7.

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 56.

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."

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]

T. D. Noe, Table of n, a(n) for n = 0..1000

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.

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]

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

G.f.: 1/(1-x)*sum(k>=1, x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 23 2003

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

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]

with(numtheory): A006218 := n->add(sigma[0](i), i=1..n);

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]

Table[Sum[DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]

FoldList[Plus, 0, Table[DivisorSigma[0, x], {x, 61}]]//Rest (much faster)

(PARI) a(n)=sum(k=1, n, n\k)

Right edge of A056535. Cf. A000005, A001659, A052511, A143236.

Row sums of triangle A003988.

A061017 is an inverse.

It appears that the partial sums give A078567. - N. J. A. Sloane (njas(AT)research.att.com), Nov 24 2008

Cf. A116477, A051731, A143724 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2008]

A161700. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

Sequence in context: A027922 A051611 A005004 this_sequence A062839 A088940 A088937

Adjacent sequences: A006215 A006216 A006217 this_sequence A006219 A006220 A006221

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Comment about explicit bounds from Chandrasekharan [From David Applegate (david(AT)research.att.com), Oct 14 2008]