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

Author

Timothy L. Tiffin, Jul 22 2016

Keywords

• nonn

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.

Examples

			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.
...      ...                 ...          ...         ...
		

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].

Crossrefs

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

Programs

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 *)

Formula

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