A275316 Average of amicable pairs (x,y), ordered by the sum x+y given in A259953.

252, 1197, 2772, 5292, 6300, 10800, 13440, 17856, 66960, 69120, 69552, 78624, 84240, 112320, 122760, 131040, 147420, 155520, 174096, 178560, 194400, 199584, 322812, 349272, 374976, 378000, 446400, 477603, 508896, 524160, 635040, 648000, 648000, 657720, 673920, 725760, 761400, 833280, 890568, 939600

Offset

1,1

Comments

Each term represents the midpoint of an interval (x,y), where x (A260086) and y (A260087) form a pair of amicable numbers (A259933). The length and radius of each interval can be found in A275469 and A275470, respectively.

This sequence is monotonic (specifically, nondecreasing), since x+y (A259953) is nondecreasing. For a nonmonotonic ordering of these averages, see A275315.

It is unknown if there exists an amicable pair where x and y have opposite parity (one is even and the other is odd). If such a pair is ever found, then the compound adjective "same-parity" will need to be added to the name of this sequence.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..142 from Timothy L. Tiffin)

VaxaSoftware, List of amicable numbers from 1 to 20,000,000 [142 pairs].

Formula

a(n) = [A260086(n) + A260087(n)]/2 = A259953(n)/2.

Example

a(  1) = (     220 +      284)/2 =      504/2 =      252.
a(  2) = (    1184 +     1210)/2 =     2394/2 =     1197.
a(  3) = (    2620 +     2924)/2 =     5544/2 =     2772.
...      ...                 ...          ...         ...
a(  9) = (   66928 +    66992)/2 =   133920/2 =    66960.
a( 10) = (   67095 +    71145)/2 =   138240/2 =    69120.
a( 11) = (   63020 +    76084)/2 =   139104/2 =    69552.
...      ...                 ...          ...         ...
a( 15) = (  122368 +   123152)/2 =   245520/2 =   122760.
a( 16) = (  122265 +   139815)/2 =   262080/2 =   131040.
a( 17) = (  141664 +   153176)/2 =   294840/2 =   147420.
...      ...                 ...          ...         ...
a( 32) = (  609928 +   686072)/2 =  1296000/2 =   648000.
a( 33) = (  643336 +   652664)/2 =  1296000/2 =   648000.
...      ...                 ...          ...         ...
a(107) = ( 9478910 + 11049730)/2 = 20528640/2 = 10264320.
a(108) = (10254970 + 10273670)/2 = 20528640/2 = 10264320.
...      ...                 ...          ...         ...
a(139) = (17754165 + 19985355)/2 = 37739520/2 = 18869760.
a(140) = (17844255 + 19895265)/2 = 37739520/2 = 18869760.
...      ...                 ...          ...         ...

Mathematica

With[{s = PositionIndex@ Array[DivisorSigma[1, #] &, 10^6]}, Flatten@ Map[Mean, Apply[Join, Map[Function[n, Select[Subsets[Lookup[s, n], {2}], Total@ # == n &]], Sort@ Select[Keys@ s, Length@ Lookup[s, #] > 1 &]]]]] (* Michael De Vlieger, Oct 22 2017 *)

Crossrefs

Cf. A260086, A260087, A259180, A259933, A259953, A275315, A275469, A275470.

Sequence in context: A250794 A245858 A200884 * A275315 A184505 A184499

Adjacent sequences: A275313 A275314 A275315 * A275317 A275318 A275319

Keyword

nonn

Author

Timothy L. Tiffin, Jul 22 2016

Status

approved