A024916 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).
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
Offset: 1
Examples
From _Omar E. Pol_, Aug 20 2021: (Start)
For n = 6 the sum of all divisors of the first six positive integers is [1] + [1 + 2] + [1 + 3] + [1 + 2 + 4] + [1 + 5] + [1 + 2 + 3 + 6] = 1 + 3 + 4 + 7 + 6 + 12 = 33, so a(6) = 33.
On the other hand the area under the Dyck path of the 6th diagram as shown below is equal to 33, so a(6) = 33.
Illustration of initial terms: _ _ _ _
_ _ _ | |_
_ _ _ | | | |_
_ _ | |_ | |_ _ | |
_ _ | |_ | | | | | |
_ | | | | | | | | | |
|_| |_ _| |_ _ _| |_ _ _ _| |_ _ _ _ _| |_ _ _ _ _ _|
.
1 4 8 15 21 33 (End)
References
- Hardy and Wright, "An introduction to the theory of numbers", Oxford University Press, fifth edition, p. 266.
Links
- Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
- Vaclav Kotesovec, Plot of (a(n) - Pi^2*n^2/12) / (n*log(n)^(2/3)) for n = 2..100000.
- P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916.
- Peter Polm, C# program for A024916.
- A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 44, Issue 12, page 607, 1964.
Crossrefs
Programs
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Haskell
a024916 n = sum $ map (\k -> k * div n k) [1..n] -- Reinhard Zumkeller, Apr 20 2015
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Magma
[(&+[DivisorSigma(1, k): k in [1..n]]): n in [1..60]]; // G. C. Greubel, Mar 15 2019
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Maple
A024916 := proc(n) add(numtheory[sigma](k),k=0..n) ; end proc: # Zerinvary Lajos, Jan 11 2009 # second Maple program: a:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+a(n-1)) end: seq(a(n), n=1..100); # Alois P. Heinz, Sep 12 2019
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Mathematica
Table[Plus @@ Flatten[Divisors[Range[n]]], {n, 50}] (* Alonso del Arte, Mar 06 2006 *) Table[Sum[n - Mod[n, m], {m, n}], {n, 50}] (* Roger L. Bagula and Gary W. Adamson, Oct 06 2006 *) a[n_] := Sum[DivisorSigma[1, k], {k, n}]; Table[a[n], {n, 51}] (* Jean-François Alcover, Dec 16 2011 *) Accumulate[DivisorSigma[1,Range[60]]] (* Harvey P. Dale, Mar 13 2014 *) -
PARI
A024916(n)=sum(k=1,n,n\k*k) \\ M. F. Hasler, Nov 22 2007
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PARI
A024916(z) = { my(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); } \\ See the link for a nicely formatted version. - P. L. Patodia (pannalal(AT)usa.net), Jan 11 2008
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PARI
A024916(n)={my(s=0,d=1,q=n);while(d
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PARI
A024916(n)={ my(s=n^2, r=sqrtint(n), nd=n, D); for(d=1, r, (1>=D=nd-nd=n\(d+1)) && (r=d-1) && break; s -= n%d*D+(D-1)*D\2*d); s - sum(d=2, n\(r+1), n%d)} \\ Slightly optimized version of Patodia's code. - M. F. Hasler, Apr 18 2015 (C#) See Polm link.
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Python
def A024916(n): return sum(k*(n//k) for k in range(1,n+1)) # Chai Wah Wu, Dec 17 2021
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Python
from math import isqrt def A024916(n): return (-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1 # Chai Wah Wu, Oct 21 2023
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Sage
[sum(sigma(k) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Mar 15 2019
Formula
From Benoit Cloitre, Apr 28 2002: (Start)
a(n) = n^2 - A004125(n).
Asymptotically a(n) = n^2*Pi^2/12 + O(n*log(n)). (End)
G.f.: (1/(1-x))*Sum_{k>=1} x^k/(1-x^k)^2. - Benoit Cloitre, Apr 23 2003
a(n) = Sum_{m=1..n} (n - (n mod m)). - Roger L. Bagula and Gary W. Adamson, Oct 06 2006
a(n) = n^2*Pi^2/12 + O(n*log(n)^(2/3)) [Walfisz]. - Charles R Greathouse IV, Jun 19 2012
From Greg Dresden, Feb 23 2020: (Start)
a(n) = A092406(n) + 8, n>3.
a(n) = A160664(n) - 1, n>0. (End)
a(n) = Sum_{k=1..n} k * A010766(n,k). - Georg Fischer, Mar 04 2022
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