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A317307
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Sum of divisors of powers of 2 and sum of divisors of even perfect numbers.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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EXAMPLE
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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 |_ _ _ _ _ _ _ _ _| | | _ _ _ _ _ _| |
. _ _| | | _ _ _ _ _ _|
. _ _| _ _| | |
. | _| _ _| |
. _| _| | _ _|
. | _| _| |
. _ _ _| | _| _|
. | _ _ _| _ _| _|
. | | | _ _|
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. | | | _ _ _|
. _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
. 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |
. | |
. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
. 63 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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Odd terms give the nonzeros terms of A000225.
Cf. A000203, A000396, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A317306.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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