A317306 Powers of 2 and even perfect numbers.

1, 2, 4, 6, 8, 16, 28, 32, 64, 128, 256, 496, 512, 1024, 2048, 4096, 8128, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33550336, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589869056, 8589934592

Offset: 1

Author

Omar E. Pol, Aug 23 2018

Keywords

• nonn
• easy

Comments

Numbers k such that the symmetric representation of sigma(k) has only one part, and apart from the central width, the rest of the widths are 1's.

Note that the above definition implies that the central width of the symmetric representation of sigma(k) is 1 or 2. For powers of 2 the central width is 1. For even perfect numbers the central width is 2 (see example).

Examples

			Illustration of initial terms:
.        _ _   _   _   _               _                       _       _
.    1  |_| | | | | | | |             | |                     | |     | |
.    2  |_ _|_| | | | | |             | |                     | |     | |
.        _ _|  _|_| | | |             | |                     | |     | |
.    4  |_ _ _|    _|_| |             | |                     | |     | |
.        _ _ _|  _|  _ _|             | |                     | |     | |
.    6  |_ _ _ _|  _|                 | |                     | |     | |
.        _ _ _ _| |                   | |                     | |     | |
.    8  |_ _ _ _ _|              _ _ _| |                     | |     | |
.                               |  _ _ _|                     | |     | |
.                              _| |                           | |     | |
.                            _|  _|                           | |     | |
.                        _ _|  _|                             | |     | |
.                       |  _ _|                               | |     | |
.                       | |                          _ _ _ _ _| |     | |
.        _ _ _ _ _ _ _ _| |                         |  _ _ _ _ _|     | |
.   16  |_ _ _ _ _ _ _ _ _|                         | |    _ _ _ _ _ _| |
.                                                _ _| |   |  _ _ _ _ _ _|
.                                            _ _|  _ _|   | |
.                                           |    _|    _ _| |
.                                          _|  _|     |  _ _|
.                                         |  _|      _| |
.                                    _ _ _| |      _|  _|
.                                   |  _ _ _|  _ _|  _|
.                                   | |       |  _ _|
.                                   | |  _ _ _| |
.                                   | | |  _ _ _|
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
.   28  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.                                       | |
.                                       | |
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.   32  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram shows the first eight terms of the sequence. The symmetric representation of sigma has only one part, and apart from the central width, the rest of the widths are 1's.
A317307(n) is the area (or the number of cells) in the n-th region of the diagram.
		

Crossrefs

Union of A000079 and A000396 assuming there are no odd perfect numbers.

Subsequence of A174973.

Cf. A249351 (the widths).

Cf. A317307(n) = sigma(a(n)).

Cf. A000203, A000225, A139256, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626.