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A317305
Sum of divisors of the n-th number whose divisors increase by a factor of 2 or less.
6
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.
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
A317305[upto_]:=Table[If[AllTrue[Map[Last[#]/First[#]&, Partition[Divisors[n], 2, 1]], #<=2&], DivisorSigma[1, n], Nothing], {n, upto}];
A317305[500] (* Paolo Xausa, Jan 12 2023 *)
KEYWORD
nonn
AUTHOR
Omar E. Pol, Aug 25 2018
STATUS
approved