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A358683 a(n) is the sum of all divisors of all positive integers k where A182986(n) < k <= prime(n), n >= 1.

%I A358683 #61 Dec 21 2022 20:47:28
%S A358683 4,4,13,20,58,42,97,59,134,259,104,342,248,140,282,498,542,230,623,
%T A358683 438,269,722,517,854,1256,646,320,672,390,730,2767,815,1348,428,2361,
%U A358683 524,1564,1553,1002,1729,1670,728,2980,702,1227,668,4125,4172,1477,790,1500,2246,986,3859,2601,2470,2630
%N A358683 a(n) is the sum of all divisors of all positive integers k where A182986(n) < k <= prime(n), n >= 1.
%C A358683 a(n) has a symmetric representation as shown in the example.
%F A358683 a(n) = Sum_{k=1+A182986(n)..A182986(n+1)} A000203(k), n >= 1.
%e A358683 a(1) = sigma(1) + sigma(2) = 1 + 3 = 4.
%e A358683 a(2) = sigma(3) = 4.
%e A358683 a(3) = sigma(4) + sigma(5) = 7 + 6 = 13.
%e A358683 a(4) = sigma(6) + sigma(7) = 12 + 8 = 20.
%e A358683 a(5) = sigma(8) + sigma(9) + sigma(10) + sigma(11) = 15 + 13 + 18 + 12 = 58.
%e A358683 a(6) = sigma(12) + sigma(13) = 28 + 14 = 42.
%e A358683 ...
%e A358683 a(40) = sigma(168) + sigma(169) + sigma(170) + sigma(171) + sigma(172) + sigma(173) = 480 + 183 + 324 + 260 + 308 + 174 = 1729.
%e A358683 Illustration of initial terms using the Dyck paths described in A237593:
%e A358683 .
%e A358683 .  n  prime(n)  a(n)            Diagram
%e A358683 .                      _ _ _ _ _ _ _ _ _ _ _ _ _
%e A358683 .                     |   | |   |   |       |   |
%e A358683 .  1     2       4    |_ _|_|   |   |       |   |
%e A358683 .  2     3       4    |_ _|  _ _|   |       |   |
%e A358683 .                     |     |    _ _|       |   |
%e A358683 .  3     5      13    |_ _ _|  _|           |   |
%e A358683 .                     |       |        _ _ _|   |
%e A358683 .  4     7      20    |_ _ _ _|      _|    _ _ _|
%e A358683 .                     |            _|     |
%e A358683 .                     |           |      _|
%e A358683 .                     |           |  _ _|
%e A358683 .  5    11      58    |_ _ _ _ _ _| |
%e A358683 .                     |             |
%e A358683 .  6    13      42    |_ _ _ _ _ _ _|
%e A358683 .
%e A358683 The diagram of a(40) = 1729 is too large to include.
%t A358683 {Total@ DivisorSigma[1, Range[2]]}~Join~Array[Total@ DivisorSigma[1, Range[Prime[# - 1] + 1, Prime[#]]] &, 56, 2] (* _Michael De Vlieger_, Nov 29 2022 *)
%o A358683 (PARI) A358683(n) = sum(k=if(1==n,1,1+prime(n-1)),prime(n),sigma(k)); \\ _Antti Karttunen_, Nov 29 2022
%Y A358683 Partial sums give A244583.
%Y A358683 Cf. A000040, A000203, A001235, A024916, A182986, A237270, A237591, A237593, A299763.
%K A358683 nonn
%O A358683 1,1
%A A358683 _Omar E. Pol_, Nov 26 2022