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A244583 a(n) = sum of all divisors of all positive integers <= prime(n).

%I A244583 #47 Oct 23 2023 10:31:02
%S A244583 4,8,21,41,99,141,238,297,431,690,794,1136,1384,1524,1806,2304,2846,
%T A244583 3076,3699,4137,4406,5128,5645,6499,7755,8401,8721,9393,9783,10513,
%U A244583 13280,14095,15443,15871,18232,18756,20320,21873,22875,24604,26274,27002,29982,30684
%N A244583 a(n) = sum of all divisors of all positive integers <= prime(n).
%C A244583 Limit_{n->oo} a(n)/prime(n)^2 = zeta(2)/2 = Pi^2/12 = A072691 = 0.82246703342.... For example, at n = 2*10^6, the ratio converges to 0.822467033... (+-2 in the last digit with increments on n of +100). If the ratio is calculated with a nonprime for the upper summation limit then the ratio runs slightly larger and converges slower. See formula section of A024916 for the general case. - _Richard R. Forberg_, Jan 04 2015
%C A244583 This is a subsequence of A024916 therefore a(n) also has a symmetric representation. For more information see A236104, A237593. - _Omar E. Pol_, Jan 05 2015
%F A244583 a(n) = A024916(A000040(n)).
%F A244583 a(n) = A001248(n) - A050482(n). - _Omar E. Pol_, Jan 05 2015
%t A244583 a244583[n_] := Sum[DivisorSigma[1, i], {i, #}] & /@ Prime[Range@n]; a244583[44] (* _Michael De Vlieger_, Jan 06 2015 *)
%o A244583 (PARI) a(n) = sum(i=1, prime(n), sigma(i)); \\ _Michel Marcus_, Sep 29 2014
%o A244583 (Python)
%o A244583 from math import isqrt
%o A244583 from sympy import prime
%o A244583 def A244583(n): return -(s:=isqrt(p:=prime(n)))**2*(s+1) + sum((q:=p//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # _Chai Wah Wu_, Oct 23 2023
%Y A244583 Cf. A000040, A000203, A024916, A236104, A237593, A237270, A237271, A244576, A244578.
%Y A244583 Partial sums of A358683.
%K A244583 nonn
%O A244583 1,1
%A A244583 _Omar E. Pol_, Jun 30 2014
%E A244583 More terms from _Michel Marcus_, Sep 29 2014