A317307 Sum of divisors of powers of 2 and sum of divisors of even perfect numbers.

1, 3, 7, 12, 15, 31, 56, 63, 127, 255, 511, 992, 1023, 2047, 4095, 8191, 16256, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67100672, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591, 17179738112, 17179869183

Offset: 1

Author

Omar E. Pol, Aug 25 2018

Keywords

• nonn
• easy

Comments

Sum of divisors of the 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. a(n) is the area (or the number of cells) of the n-th region of the diagram:
.        _ _   _   _   _               _                       _       _
.   1   |_| | | | | | | |             | |                     | |     | |
.   3   |_ _|_| | | | | |             | |                     | |     | |
.        _ _|  _|_| | | |             | |                     | |     | |
.   7   |_ _ _|    _|_| |             | |                     | |     | |
.        _ _ _|  _|  _ _|             | |                     | |     | |
.  12   |_ _ _ _|  _|                 | |                     | |     | |
.        _ _ _ _| |                   | |                     | |     | |
.  15   |_ _ _ _ _|              _ _ _| |                     | |     | |
.                               |  _ _ _|                     | |     | |
.                              _| |                           | |     | |
.                            _|  _|                           | |     | |
.                        _ _|  _|                             | |     | |
.                       |  _ _|                               | |     | |
.                       | |                          _ _ _ _ _| |     | |
.        _ _ _ _ _ _ _ _| |                         |  _ _ _ _ _|     | |
.  31   |_ _ _ _ _ _ _ _ _|                         | |    _ _ _ _ _ _| |
.                                                _ _| |   |  _ _ _ _ _ _|
.                                            _ _|  _ _|   | |
.                                           |    _|    _ _| |
.                                          _|  _|     |  _ _|
.                                         |  _|      _| |
.                                    _ _ _| |      _|  _|
.                                   |  _ _ _|  _ _|  _|
.                                   | |       |  _ _|
.                                   | |  _ _ _| |
.                                   | | |  _ _ _|
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
.   56  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.                                       | |
.                                       | |
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.   63  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram shows the first eight terms of the sequence. The symmetric representation of sigma of the numbers A317306: 1, 2, 4, 6, 8, 16, 28, 32, ..., has only one part, and apart from the central width, the rest of the widths are 1's.
		

Crossrefs

Union of nonzero terms of A000225 and A139256.

Odd terms give the nonzeros terms of A000225.

Even terms give A139256.

Subsequence of A317305.

Cf. A249351 (the widths).

Cf. A000203, A000396, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A317306.

Programs

Mathematica

DivisorSigma[1, #] &@ Union[2^Range[0, Floor@ Log2@ Last@ #], #] &@ Array[2^(# - 1) (2^# - 1) &@ MersennePrimeExponent@ # &, 7] (* Michael De Vlieger, Aug 25 2018, after Robert G. Wilson v at A000396 *)

Formula

a(n) = A000203(A317306(n)).