A317305 Sum of divisors of the n-th number whose divisors increase by a factor of 2 or less.

1, 3, 7, 12, 15, 28, 31, 39, 42, 60, 56, 72, 63, 91, 90, 96, 124, 120, 120, 168, 127, 144, 195, 186, 224, 180, 234, 252, 217, 210, 280, 248, 360, 312, 255, 336, 336, 403, 372, 392, 378, 363, 480, 372, 546, 508, 399, 468, 465, 504, 434, 576, 600, 504, 504, 560, 546, 744, 728, 511

Offset: 1

Author

Omar E. Pol, Aug 25 2018

Keywords

• nonn

Comments

Also consider the n-th number k with the property that the symmetric representation of sigma(k) has only one part. a(n) is the area of the diagram (see the example). For more information see A237593 and its related sequences.

Examples

			Illustration of initial terms (n = 1..13):
.
  a(n)
        _ _   _   _   _       _       _   _   _       _       _   _   _
   1   |_| | | | | | | |     | |     | | | | | |     | |     | | | | | |
   3   |_ _|_| | | | | |     | |     | | | | | |     | |     | | | | | |
        _ _|  _|_| | | |     | |     | | | | | |     | |     | | | | | |
   7   |_ _ _|    _|_| |     | |     | | | | | |     | |     | | | | | |
        _ _ _|  _|  _ _|     | |     | | | | | |     | |     | | | | | |
  12   |_ _ _ _|  _|    _ _ _| |     | | | | | |     | |     | | | | | |
        _ _ _ _| |    _|    _ _|     | | | | | |     | |     | | | | | |
  15   |_ _ _ _ _|  _|     |    _ _ _| | | | | |     | |     | | | | | |
                   |      _|   |  _ _ _|_| | | |     | |     | | | | | |
                   |  _ _|    _| |    _ _ _|_| |     | |     | | | | | |
        _ _ _ _ _ _| |      _|  _|   |  _ _ _ _|     | |     | | | | | |
  28   |_ _ _ _ _ _ _|  _ _|  _|  _ _| |    _ _ _ _ _| |     | | | | | |
                       |  _ _|  _|    _|   |    _ _ _ _|     | | | | | |
                       | |     |     |  _ _|   |    _ _ _ _ _| | | | | |
        _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |   |  _ _ _ _ _|_| | | |
  31   |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|   | |    _ _ _ _ _|_| |
        _ _ _ _ _ _ _ _ _| | |     |      _|    _ _| |   |  _ _ _ _ _ _|
  39   |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _ _|  _ _|   | |
        _ _ _ _ _ _ _ _ _ _| | |       |   |    _|    _ _| |
  42   |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|  _|     |  _ _|
                               | |       |  _|      _| |
                               | |  _ _ _| |      _|  _|
        _ _ _ _ _ _ _ _ _ _ _ _| | |  _ _ _|  _ _|  _|
  60   |_ _ _ _ _ _ _ _ _ _ _ _ _| | |       |  _ _|
                                   | |  _ _ _| |
                                   | | |  _ _ _|
        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
  56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
  72   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  63   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The length of the largest Dyck path of the n-th diagram equals A047836(n).
The semilength equals A174973(n).
a(n) is the area of the n-th diagram.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sigma(n)

Crossrefs

A317307 is a subsequence.

Cf. A174973.

Cf. A000203, A047836, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A361208.

Programs

Mathematica

A317305[upto_]:=Table[If[AllTrue[Map[Last[#]/First[#]&,Partition[Divisors[n],2,1]],#<=2&],DivisorSigma[1,n],Nothing],{n,upto}];
A317305[500] (* Paolo Xausa, Jan 12 2023 *)

Formula

a(n) = A000203(A174973(n)).