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Search: id:A006218
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| A006218 |
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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.
(Formerly M2432)
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+0
65
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| 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, 87, 91, 95, 101, 103, 111, 113, 119, 123, 127, 131, 140, 142, 146, 150, 158, 160, 168, 170, 176, 182, 186, 188, 198, 201, 207, 211, 217, 219, 227, 231, 239, 243, 247, 249
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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
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REFERENCES
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K. Chandrasekharan, (1970): Arithmetical Functions; Chapter VIII, pp. 194-228. Springer-Verlag, Berlin.
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.
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."
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
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FORMULA
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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). 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.
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)lhsystems.com), May 10 2006
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MAPLE
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with(numtheory): A006218 := n->add(sigma[0](i), i=1..n);
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MATHEMATICA
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Table[Sum[DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]
FoldList[Plus, 0, Table[DivisorSigma[0, x], {x, 61}]]//Rest (much faster)
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PROGRAM
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(PARI) a(n)=sum(k=1, n, n\k)
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CROSSREFS
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Right edge of A056535. Cf. A000005, A001659.
Row sums of triangle A003988.
Sequence in context: A027922 A051611 A005004 this_sequence A062839 A088940 A088937
Adjacent sequences: A006215 A006216 A006217 this_sequence A006219 A006220 A006221
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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