A067692 a(n) = Sum_{0 < d <= t <= n, d|n, t|n} d*t.

1, 7, 13, 35, 31, 97, 57, 155, 130, 227, 133, 497, 183, 413, 418, 651, 307, 988, 381, 1155, 762, 953, 553, 2225, 806, 1307, 1210, 2093, 871, 3242, 993, 2667, 1762, 2183, 1802, 5096, 1407, 2705, 2418, 5155, 1723, 5858

Offset: 1

Author

Reinhard Zumkeller, Feb 04 2002

Keywords

• nonn
• easy

Comments

Total area of all s X t rectangles, where the (s,t) are the pairs of divisors of n such that 1 <= s <= t. For example, when n = 4, the rectangles are 1 X 1, 1 X 2, 1 X 4, 2 X 2, 2 X 4, and 4 X 4, whose total area is a(4) = 1*1 + 1*2 + 1*4 + 2*2 + 2*4 + 4*4 = 35. - Wesley Ivan Hurt, Nov 15 2021

Examples

			a(6) = 1*1+1*2+1*3+1*6+2*2+2*3+2*6+3*3+3*6+6*6 = 1+2+3+6+4+6+12+9+18+36 = 97.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000203, A001157, A002117, A060800.

Programs

Mathematica

Table[(DivisorSigma[1,n]^2+DivisorSigma[2,n])/2,{n,50}] (* Harvey P. Dale, Jan 31 2015 *)

PARI

a(n)=my(D=sigma(n));sumdiv(n,t,D-=t;t*(D+t)) \\ Charles R Greathouse IV, Aug 21 2011

PARI

a(n)=(sigma(n)^2+sigma(n,2))/2 \\ Charles R Greathouse IV, Aug 21 2011

Formula

a(n) = (1/2)*(sigma_1(n)^2 + sigma_2(n)), cf. A000203, A001157.

For p prime: a(p) = 1 + p + p^2, a(A000040(k)) = A060800(k).

Sum_{k=1..n} a(k) = (7/12)*zeta(3) * n^3 + O(n^2*log(n)^2). - Amiram Eldar, Dec 15 2023