A066839 a(n) = sum of positive divisors k of n with k <= sqrt(n).

1, 1, 1, 3, 1, 3, 1, 3, 4, 3, 1, 6, 1, 3, 4, 7, 1, 6, 1, 7, 4, 3, 1, 10, 6, 3, 4, 7, 1, 11, 1, 7, 4, 3, 6, 16, 1, 3, 4, 12, 1, 12, 1, 7, 9, 3, 1, 16, 8, 8, 4, 7, 1, 12, 6, 14, 4, 3, 1, 21, 1, 3, 11, 15, 6, 12, 1, 7, 4, 15, 1, 24, 1, 3, 9, 7, 8, 12, 1, 20, 13, 3, 1, 23, 6, 3, 4, 15, 1, 26, 8, 7, 4, 3

Offset: 1

Author

Leroy Quet, Jan 20 2002

Keywords

• nonn

Comments

Row sums of the table in A161906. - Reinhard Zumkeller, Mar 08 2013

Conjecture: a(n) is the total number of parts in all partitions of n into consecutive parts that differ by 2. - Omar E. Pol, May 03 2020. This conjecture is true (the g.f. for these partitions agrees with the g.f. given below by Michael Somos). - N. J. A. Sloane, Dec 02 2020

Column 2 of A334466. - Omar E. Pol, Dec 03 2020

Examples

			a(9) = 4 = 1 + 3 because 1 and 3 are the positive divisors of 9 that are <= sqrt(9).
a(20) = 7: the divisors of 20 are 1, 2, 4, 5, 10 and 20. a(20) = 1 + 2 + 4 = 7.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
• Douglas E. Iannucci, On sums of the small divisors of a natural number, arXiv:1910.11835 [math.NT], 2019.

Crossrefs

Cf. A000203, A070038, A038548, A072499, A334466.

Programs

Haskell

a066839 = sum . a161906_row  -- Reinhard Zumkeller, Mar 08 2013

Maple

with(numtheory):for n from 1 to 200 do c[n] := 0:d := divisors(n):for i from 1 to nops(d) do if d[i]<=n^.5+10^(-10) then c[n] := c[n]+d[i]:fi:od:od:seq(c[i],i=1..200);
# alternative
seq(add(d, d in select(x->x^2<=n, numtheory[divisors](n))), n=1..100); # Ridouane Oudra, Jun 24 2025

Mathematica

f[n_] := Plus @@ Select[ Divisors@n, # <= Sqrt@n &]; Array[f, 94] (* Robert G. Wilson v, Mar 04 2010 *)
Table[Sum[If[n > k*(k-1), k, 0], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Oct 22 2024 *)

PARI

a(n)=sumdiv(n,d, (d^2<=n)*d) /* Michael Somos, Nov 19 2005 */

PARI

{ for (n=1, 1000, d=divisors(n); s=sum(k=1, ceil(length(d)/2), d[k]); write("b066839.txt", n, " ", s) ) } \\ Harry J. Smith, Mar 31 2010

Python

from itertools import takewhile
from sympy import divisors
def A066839(n): return sum(takewhile(lambda x:x**2<=n,divisors(n))) # Chai Wah Wu, Dec 19 2023

Sage

[sum(k for k in divisors(n) if k^2<=n) for n in (1..94)] # Giuseppe Coppoletta, Jan 21 2015

Formula

G.f.: Sum_{k>0} k*x^(k^2)/(1-x^k). - Michael Somos, Nov 19 2005

a(n) = Sum_{i=1..floor(sqrt(n))} (-(n mod i) + (n-1) mod i + 1). - José de Jesús Camacho Medina, Feb 21 2021

a(p^(2k+1)) = a(p^(2k)) = (p^(k+1)-1)/(p-1) = A000203(p^k) for k>=0 and p prime. - Chai Wah Wu, Dec 23 2023

Sum_{k=1..n} a(k) ~ 2 * n^(3/2) / 3 [Iannucci, 2019]. - Vaclav Kotesovec, Oct 23 2024

a(n) = A070039(n) + A037213(n). - Ridouane Oudra, Jun 24 2025

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2002