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

4, 4, 13, 20, 58, 42, 97, 59, 134, 259, 104, 342, 248, 140, 282, 498, 542, 230, 623, 438, 269, 722, 517, 854, 1256, 646, 320, 672, 390, 730, 2767, 815, 1348, 428, 2361, 524, 1564, 1553, 1002, 1729, 1670, 728, 2980, 702, 1227, 668, 4125, 4172, 1477, 790, 1500, 2246, 986, 3859, 2601, 2470, 2630

Offset: 1

Author

Omar E. Pol, Nov 26 2022

Keywords

• nonn

Comments

a(n) has a symmetric representation as shown in the example.

Examples

			a(1) = sigma(1) + sigma(2) = 1 + 3 = 4.
a(2) = sigma(3) = 4.
a(3) = sigma(4) + sigma(5) = 7 + 6 = 13.
a(4) = sigma(6) + sigma(7) = 12 + 8 = 20.
a(5) = sigma(8) + sigma(9) + sigma(10) + sigma(11) = 15 + 13 + 18 + 12 = 58.
a(6) = sigma(12) + sigma(13) = 28 + 14 = 42.
...
a(40) = sigma(168) + sigma(169) + sigma(170) + sigma(171) + sigma(172) + sigma(173) = 480 + 183 + 324 + 260 + 308 + 174 = 1729.
Illustration of initial terms using the Dyck paths described in A237593:
.
.  n  prime(n)  a(n)            Diagram
.                      _ _ _ _ _ _ _ _ _ _ _ _ _
.                     |   | |   |   |       |   |
.  1     2       4    |_ _|_|   |   |       |   |
.  2     3       4    |_ _|  _ _|   |       |   |
.                     |     |    _ _|       |   |
.  3     5      13    |_ _ _|  _|           |   |
.                     |       |        _ _ _|   |
.  4     7      20    |_ _ _ _|      _|    _ _ _|
.                     |            _|     |
.                     |           |      _|
.                     |           |  _ _|
.  5    11      58    |_ _ _ _ _ _| |
.                     |             |
.  6    13      42    |_ _ _ _ _ _ _|
.
The diagram of a(40) = 1729 is too large to include.
		

Crossrefs

Partial sums give A244583.

Cf. A000040, A000203, A001235, A024916, A182986, A237270, A237591, A237593, A299763.

Programs

Mathematica

{Total@ DivisorSigma[1, Range[2]]}~Join~Array[Total@ DivisorSigma[1, Range[Prime[# - 1] + 1, Prime[#]]] &, 56, 2] (* Michael De Vlieger, Nov 29 2022 *)

PARI

A358683(n) = sum(k=if(1==n,1,1+prime(n-1)),prime(n),sigma(k)); \\ Antti Karttunen, Nov 29 2022

Formula

a(n) = Sum_{k=1+A182986(n)..A182986(n+1)} A000203(k), n >= 1.