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Search: id:A024916
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| A024916 |
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a(n) = sum_{k=1..n} k*floor(n/k); also sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203). |
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38
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| 1, 4, 8, 15, 21, 33, 41, 56, 69, 87, 99, 127, 141, 165, 189, 220, 238, 277, 297, 339, 371, 407, 431, 491, 522, 564, 604, 660, 690, 762, 794, 857, 905, 959, 1007, 1098, 1136, 1196, 1252, 1342, 1384, 1480, 1524, 1608, 1686, 1758, 1806, 1930, 1987, 2080, 2152
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(10^4) = 82256014, a(10^5) = 8224740835, a(10^6) = 822468118437, a(10^7) = 82246711794796; see A072692. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Nov 22 2007
Equals row sums of triangle A158903 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 29 2009]
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REFERENCES
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For the asymptotic formula see Theorem 324 in Hardy and Wright "An introduction to the theory of numbers", Oxford university press, fifth edition, p. 266
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916
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FORMULA
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a(n)=n^2-A004125(n); asymptotically a(n)=n^2*Pi^2/12+O(n*Log(n)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 28 2002
G.f.: 1/(1-x)*sum(k>=1, x^k/(1-x^k)^2). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 23 2003
a(n) = Sum[n - Mod[n, m], {m, 1, n}] - Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 06 2006
Row sums of triangle A130541. E.g. a(5) = 15 = (10 + 3 + 1 + 1), sum of row 4 terms of triangle A130541. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 03 2007
Row sums of triangle A134867 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 14 2007
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MAPLE
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with(numtheory):seq(sum(sigma(k), k=0..n), n=1..49); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 11 2009]
with(numtheory):a[1]:=1: for n from 2 to 51 do a[n]:=a[n-1]+sigma(n) od: seq(a[n], n=1..51); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2009]
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MATHEMATICA
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Table[Plus @@ Flatten[Divisors[Range[n]]], {n, 50}] - Alonso Delarte (alonso.delarte(AT)gmail.com), Mar 06 2006
Table[Sum[n - Mod[n, m], {m, 1, n}], {n, 1, 50}] - Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 06 2006
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PROGRAM
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(PARI) A024916(n)=sum(k=1, n, n\k*k) - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Nov 22 2007
(PARI) A024916(z) = { local(s, u, d, n, a, p); s = z*z; u = sqrtint(z); p = 2; for(d=1, u, n = z\d - z\(d+1); if(n<=1, p=d; break(), a = z%d; s -= (2*a+(n-1)*d)*n/2); ); u = z\p; for(d=2, u, s -= z%d); return(s); } - P L Patodia (pannalal(AT)usa.net), Jan 11 2008
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CROSSREFS
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Cf. A056550, A104471(2*n-1, n).
Cf. A123229, A130541, A000217, A134867, A072692.
A158903 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 29 2009]
Sequence in context: A136403 A071422 A113902 this_sequence A102216 A001182 A122247
Adjacent sequences: A024913 A024914 A024915 this_sequence A024917 A024918 A024919
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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