%I #34 Aug 26 2018 12:35:28
%S 1,3,7,12,15,31,56,63,127,255,511,992,1023,2047,4095,8191,16256,16383,
%T 32767,65535,131071,262143,524287,1048575,2097151,4194303,8388607,
%U 16777215,33554431,67100672,67108863,134217727,268435455,536870911,1073741823,2147483647,4294967295,8589934591,17179738112,17179869183
%N Sum of divisors of powers of 2 and sum of divisors of even perfect numbers.
%C 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.
%C 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).
%F a(n) = A000203(A317306(n)).
%e Illustration of initial terms. a(n) is the area (or the number of cells) of the n-th region of the diagram:
%e . _ _ _ _ _ _ _ _
%e . 1 |_| | | | | | | | | | | | | |
%e . 3 |_ _|_| | | | | | | | | | | |
%e . _ _| _|_| | | | | | | | | |
%e . 7 |_ _ _| _|_| | | | | | | |
%e . _ _ _| _| _ _| | | | | | |
%e . 12 |_ _ _ _| _| | | | | | |
%e . _ _ _ _| | | | | | | |
%e . 15 |_ _ _ _ _| _ _ _| | | | | |
%e . | _ _ _| | | | |
%e . _| | | | | |
%e . _| _| | | | |
%e . _ _| _| | | | |
%e . | _ _| | | | |
%e . | | _ _ _ _ _| | | |
%e . _ _ _ _ _ _ _ _| | | _ _ _ _ _| | |
%e . 31 |_ _ _ _ _ _ _ _ _| | | _ _ _ _ _ _| |
%e . _ _| | | _ _ _ _ _ _|
%e . _ _| _ _| | |
%e . | _| _ _| |
%e . _| _| | _ _|
%e . | _| _| |
%e . _ _ _| | _| _|
%e . | _ _ _| _ _| _|
%e . | | | _ _|
%e . | | _ _ _| |
%e . | | | _ _ _|
%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
%e . 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e . | |
%e . | |
%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e . 63 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e .
%e 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.
%t 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 *)
%Y Union of nonzero terms of A000225 and A139256.
%Y Odd terms give the nonzeros terms of A000225.
%Y Even terms give A139256.
%Y Subsequence of A317305.
%Y Cf. A249351 (the widths).
%Y Cf. A000203, A000396, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A317306.
%K nonn,easy
%O 1,2
%A _Omar E. Pol_, Aug 25 2018