OFFSET
1,2
COMMENTS
Row sums of triangleA128489.E.g., a(5) = 15 = (10 + 3 + 1 + 1), sum of row 4 terms of triangleA128489.-Gary W. Adamson,Jun 03 2007
Row sums of triangleA134867.-Gary W. Adamson,Nov 14 2007
a(10^4) = 82256014, a(10^5) = 8224740835, a(10^6) = 822468118437, a(10^7) = 82246711794796; seeA072692.-M. F. Hasler,Nov 22 2007
Equals row sums of triangleA158905.-Gary W. Adamson,Mar 29 2009
n is prime if and only if a(n) - a(n-1) - 1 = n. -Omar E. Pol,Dec 31 2012
Also the alternating row sums ofA236104.-Omar E. Pol,Jul 21 2014
a(n) is also the total number of parts in all partitions of the positive integers <= n into equal parts. -Omar E. Pol,Apr 30 2017
a(n) is also the total area of the terraces of the stepped pyramid with n levels described inA245092.-Omar E. Pol,Nov 04 2017
a(n) is also the area under the Dyck path described in the n-th row ofA237593(see example). -Omar E. Pol,Sep 17 2018
FromOmar E. Pol,Feb 17 2020: (Start)
a(n) is also the number of cubic cells (or cubes) in the n-th level starting from the top of the stepped pyramid described inA245092.-Omar E. Pol,Jan 12 2022
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.
FORMULA
FromBenoit 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. BagulaandGary 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
FromGreg 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
EXAMPLE
FromOmar 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)
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
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. BagulaandGary 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 *)
PROG
(PARI)A024916(n)=sum(k=1, n, n\k*k) \\M. F. Hasler,Nov 22 2007
(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
(PARI)A024916(n)={my(s=0, d=1, q=n); while(d<q, s+=q*(q+1+2*d)\2; d++; q=n\d; ); return(s-d*(d-1)\2*d+q*(q+1)\2); } \\Peter Polm,Aug 18 2014
(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.
(Haskell)
a024916 n = sum $ map (\k -> k * div n k) [1..n]
--Reinhard Zumkeller,Apr 20 2015
(Magma) [(&+[DivisorSigma(1, k): k in [1..n]]): n in [1..60]]; //G. C. Greubel,Mar 15 2019
(Sage) [sum(sigma(k) for k in (1..n)) for n in (1..60)] #G. C. Greubel,Mar 15 2019
(Python)
defA024916(n): return sum(k*(n//k) for k in range(1, n+1)) #Chai Wah Wu,Dec 17 2021
(Python)
from math import isqrt
defA024916(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
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
STATUS
approved