A067692 - OEIS

%I #29 Dec 15 2023 06:19:21

%S 1,7,13,35,31,97,57,155,130,227,133,497,183,413,418,651,307,988,381,

%T 1155,762,953,553,2225,806,1307,1210,2093,871,3242,993,2667,1762,2183,

%U 1802,5096,1407,2705,2418,5155,1723,5858

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

%C 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

%H Charles R Greathouse IV, <a href="/A067692/b067692.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%e 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.

%t Table[(DivisorSigma[1,n]^2+DivisorSigma[2,n])/2,{n,50}] (* _Harvey P. Dale_, Jan 31 2015 *)

%o (PARI) a(n)=my(D=sigma(n));sumdiv(n,t,D-=t;t*(D+t)) \\ _Charles R Greathouse IV_, Aug 21 2011

%o (PARI) a(n)=(sigma(n)^2+sigma(n,2))/2 \\ _Charles R Greathouse IV_, Aug 21 2011

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

%K nonn,easy

%O 1,2

%A _Reinhard Zumkeller_, Feb 04 2002