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%I #75 Nov 27 2023 03:04:18
%S 0,0,0,2,2,7,7,13,16,23,23,38,38,47,55,69,69,89,89,110,120,133,133,
%T 168,173,188,200,227,227,268,268,298,312,331,343,397,397,418,434,483,
%U 483,536,536,575,607,632,632,707,714,756,776,821,821,886,902
%N Sum of all proper divisors of all positive integers <= n.
%C The proper divisors of n are all divisors except 1 and n itself. Therefore noncomposite numbers have no proper divisors.
%C For the sum of all aliquot divisors of all positive integers <= n see A153485.
%C For the sum all divisors of all positive integers <= n see A024916.
%C a(n) = a(n - 1) if and only if n is prime.
%C For n >= 3 a(n) equals the area of an arrowhead-shaped polygon formed by two zig-zag paths and the Dyck path described in the n-th row of A237593 as shown in the Links section. Note that there is a similar diagram of A153485(n) in A153485. - _Omar E. Pol_, Jun 14 2022
%H Paolo Xausa, <a href="/A244049/b244049.txt">Table of n, a(n) for n = 1..10000</a>
%H Omar E. Pol, <a href="/A244049/a244049.jpg">Illustration of initial terms with arrowhead-shaped polygons</a>.
%F a(n) = A024916(n) - A034856(n).
%F a(n) = A153485(n) - n + 1.
%F G.f.: (1/(1 - x))*Sum_{k>=2} k*x^(2*k)/(1 - x^k). - _Ilya Gutkovskiy_, Jan 22 2017
%F a(n) = A161680(n-1) - A004125(n). - _Omar E. Pol_, Mar 25 2021
%F a(n) = A000290(n) - A034856(n) - A004125(n). - _Omar E. Pol_, Mar 26 2021
%F a(n) = c * n^2 + O(n*log(n)), where c = Pi^2/12 - 1/2 = 0.322467... . - _Amiram Eldar_, Nov 27 2023
%e a(4) = 2 because the only proper divisor of 4 is 2 and the previous n contributed no proper divisors to the sum.
%e a(5) = 2 because 5 is prime and contributes no proper divisors to the sum.
%e a(6) = 7 because the proper divisors of 6 are 2 and 3, which add up to 5, and a(5) + 5 = 2 + 5 = 7.
%t propDivsRunSum[1] := 0; propDivsRunSum[n_] := propDivsRunSum[n] = propDivsRunSum[n - 1] + (Plus@@Divisors[n]) - (n + 1); Table[propDivsRunSum[n], {n, 60}] (* _Alonso del Arte_, Jun 30 2014 *)
%t Accumulate[Join[{0},Table[Total[Most[Divisors[n]]]-1,{n,2,60}]]] (* _Harvey P. Dale_, Aug 12 2016 *)
%t Accumulate[Join[{0}, Table[DivisorSigma[1, n] - n - 1, {n, 2, 55}]]] (* _Amiram Eldar_, Jun 18 2022 *)
%o (PARI) a(n) = sum(k=2, n, sigma(k)-k-1); \\ _Michel Marcus_, Mar 30 2021
%o (Python)
%o from math import isqrt
%o def A244049(n): return ((-n*(n+3)-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1)+1 # _Chai Wah Wu_, Oct 21 2023
%Y Partial sums of A048050.
%Y Cf. A000040, A000203, A000290, A001065, A004125, A008578, A024916, A027750, A034856, A153485, A161680, A237591, A237593, A245092, A262626.
%K nonn,easy
%O 1,4
%A _Omar E. Pol_, Jun 24 2014