A064949 a(n) = Sum_{i|n, j|n} min(i,j).

1, 5, 6, 15, 8, 32, 10, 37, 23, 42, 14, 100, 16, 52, 52, 83, 20, 125, 22, 132, 64, 72, 26, 252, 45, 82, 76, 162, 32, 286, 34, 177, 88, 102, 88, 397, 40, 112, 100, 336, 44, 352, 46, 222, 208, 132, 50, 572, 75, 239, 124, 252, 56, 416, 120, 414, 136, 162, 62, 916, 64

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Harry J. Smith)

Formula

a(n) = Sum_{i=1..tau(n)} (2*tau(n)-2*i+1)*d_i, where {d_i}, i=1..tau(n), is increasing sequence of divisors of n.

a(n) = Sum_{i=1..n} A135539(n,i)^2. - Ridouane Oudra, Oct 25 2021

a(n) = A000203(n) * (2*A000005(n)+1) - 2*A064944(n). - Amiram Eldar, Jan 13 2025

From Ridouane Oudra, Aug 13 2025: (Start)

a(n) = A064945(n) + A064947(n).

a(n) = 2*A064947(n) + A000203(n).

a(n) = 2*A064945(n) - A000203(n).

a(n) = 2*A064840(n) - A064948(n). (End)

Example

a(6) = dot_product(7,5,3,1)*(1,2,3,6) = 7*1 + 5*2 + 3*3 + 1*6 = 32.

Maple

with(numtheory): seq(add((2*tau(n)-2*i+1)*sort(convert(divisors(n), 'list'))[i], i=1..tau(n)), n=1..200);

Mathematica

Array[Function[{t, d}, Total@ MapIndexed[#1 (2 t - 2 First[#2] + 1) &, d]] @@ {DivisorSigma[0, #], Divisors[#]} &, 61] (* Michael De Vlieger, Oct 25 2021 *)

Prog

(PARI) a(n) = { my(d=divisors(n), t=length(d)); sum(i=1, t, (2*t - 2*i + 1)*d[i]) } \\ Harry J. Smith, Oct 01 2009
(PARI) A064949(n) = { my(i=0, u=numdiv(n)); sumdiv(n, d, i++; (((2*u)-(2*i))+1)*d); }; \\ Antti Karttunen, Nov 14 2021

Crossrefs

Cf. A000005, A000203, A064944, A064945, A135539, A064840.

Sequence in context: A145491 A282466 A060724 * A160109 A351172 A191213

Adjacent sequences: A064946 A064947 A064948 * A064950 A064951 A064952

Keyword

nonn

Author

Vladeta Jovovic, Oct 28 2001

Status

approved