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

1,2

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.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sigma(n)

Formula

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

Example

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.

Mathematica

DivisorSigma[1, Select[Range[500], Length[Split[Divisors[#], #2 <= 2*# &]] == 1 &]] (* Paolo Xausa, Oct 23 2025 *)

Prog

(Python)
from itertools import count, islice
from sympy import divisors
def A317305_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue, 1)):
        ds, c = divisors(k), k
        for i in range(len(ds)-1):
            c += ds[i]
            if (ds[i]<<1)<ds[i+1]:
                break
        else:
            yield c
A317305_list = list(islice(A317305_gen(), 30)) # Chai Wah Wu, Oct 22 2025

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.

Sequence in context: A075895 A033015 A225574 * A356425 A096998 A356353

Adjacent sequences: A317302 A317303 A317304 * A317306 A317307 A317308

Keyword

nonn

Author

Omar E. Pol, Aug 25 2018

Status

approved