cp's OEIS Frontend

A243980 Four times the sum of all divisors of all positive integers <= n.

%I A243980 #33 Oct 22 2023 00:36:48
%S A243980 4,16,32,60,84,132,164,224,276,348,396,508,564,660,756,880,952,1108,
%T A243980 1188,1356,1484,1628,1724,1964,2088,2256,2416,2640,2760,3048,3176,
%U A243980 3428,3620,3836,4028,4392,4544,4784,5008,5368,5536,5920,6096,6432,6744,7032,7224,7720
%N A243980 Four times the sum of all divisors of all positive integers <= n.
%C A243980 Also number of "ON" cells at n-th stage in a structure which looks like a simple 2-dimensional cellular automaton (see example). The structure is formed by the reflection on the four quadrants from the diagram of the symmetry of sigma in the first quadrant after n-th stage, hence the area in each quadrant equals the area of each wedge and equals A024916(n); the sum of all divisors of all positive integers <= n. For more information about the diagram see A237593 and A237270.
%H A243980 Indranil Ghosh, <a href="/A243980/b243980.txt">Table of n, a(n) for n = 1..7342</a>
%F A243980 a(n) = A016742(n) - 4*A004125(n) = 4*A024916(n).
%F A243980 a(n) = 2*(A006218(n) + A222548(n)) = 2*A327329(n). - _Omar E. Pol_, Sep 25 2019
%e A243980 Illustration of the structure after 16 stages (contains 880 ON cells):
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%t A243980 Accumulate[4*DivisorSigma[1,Range[50]]] (* _Harvey P. Dale_, May 13 2018 *)
%o A243980 (Python)
%o A243980 from math import isqrt
%o A243980 def A243980(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
%Y A243980 Partial sums of A239050.
%Y A243980 Partial sums give A244050.
%Y A243980 Cf. A000203, A000290, A004125, A016742, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934.
%K A243980 nonn
%O A243980 1,1
%A A243980 _Omar E. Pol_, Jun 18 2014