cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Timothy L. Tiffin, Jul 22 2016

Keywords

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

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.

A259933 Amicable pairs (x < y) ordered by nondecreasing sum (x + y) and then by increasing x.

Original entry on oeis.org

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 66928, 66992, 67095, 71145, 63020, 76084, 69615, 87633, 79750, 88730, 100485, 124155, 122368, 123152, 122265, 139815, 141664, 153176, 142310, 168730, 171856, 176336, 176272, 180848, 185368, 203432, 196724, 202444, 280540, 365084, 308620, 389924
Offset: 1

Views

Author

Omar E. Pol, Jul 09 2015

Keywords

Comments

A pair of numbers x and y is called amicable if the sum of the proper divisors (or aliquot parts) of either one is equal to the other.
By definition a property of the amicable pair (x, y) is that x + y = sigma(x) = sigma(y).
The amicable pairs (x < y) are adjacent to each other in the list.
Also A260086 and A260087 interleaved.
Another version (A259180) lists the amicable pairs (x < y) ordered by increasing x.
Amicable numbers A063990 are the terms of this sequence in increasing order.
First differs from both A063990 and A259180 at a(17).

Examples

			-----------------------------------
       Amicable pair         Sum
          x      y          x + y
-----------------------------------
n     A260086 A260087      A259953
-----------------------------------
1        220     284          504
2       1184    1210         2394
3       2620    2924         5544
4       5020    5564        10584
5       6232    6368        12600
6      10744   10856        21600
7      12285   14595        26880
8      17296   18416        35712
9      66928   66992       133920
10     67095   71145       138240
11     63020   76084       139104
12     69615   87633       157248
...      ...     ...          ...
32    609928  686072      1296000
33    643336  652664      1296000
...
The sum of the proper divisors (or aliquot parts) of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. On the other hand the sum of the proper divisors (or aliquot parts) of 284 is 1 + 2 + 4 + 71 + 142 = 220. Note that 220 + 284 = sigma(220) = sigma(284) = 504. The sum 220 + 284 = 504 is the smallest sum of an amicable pair, so a(1) = 220 and a(2) = 284.
Note that some pairs (x, y) share the same sum (x + y), for example: (609928 + 686072) = (643336 + 652664) = sigma(609928) = sigma(686072) = sigma(643336) = sigma(652664) = 1296000, thus in the list first appears the pair (609928, 686072) and then (643336, 652664) because 609928 < 643336.

Links

Crossrefs

Cf. A000203, A063990, A259180, A259953, A260086, A260087.

Formula

a(2n-1) + a(2n) = A000203(a(2n-1)) = A000203(a(2n)) = A259953(n).

A260086 Smaller of amicable pair (x, y) as they are listed in A259933.

Original entry on oeis.org

220, 1184, 2620, 5020, 6232, 10744, 12285, 17296, 66928, 67095, 63020, 69615, 79750, 100485, 122368, 122265, 141664, 142310, 171856, 176272, 185368, 196724, 280540, 308620, 319950, 356408, 437456, 469028, 503056, 522405, 600392, 609928, 643336, 624184, 635624, 667964, 726104, 802725, 879712, 898216, 998104, 947835
Offset: 1

Views

Author

Omar E. Pol, Jul 15 2015

Keywords

Comments

Another version of A002025.
First differs from A002025 at a(9).

Links

Crossrefs

Cf. A002025, A063990, A259180, A259933, A259953, A260087.

Formula

a(n) = A259933(2n-1) = A259953(n) - A259933(2n) = A259953(n) - A260087(n).

A260087 Larger of amicable pair (x, y) as they are listed in A259933.

Original entry on oeis.org

284, 1210, 2924, 5564, 6368, 10856, 14595, 18416, 66992, 71145, 76084, 87633, 88730, 124155, 123152, 139815, 153176, 168730, 176336, 180848, 203432, 202444, 365084, 389924, 430402, 399592, 455344, 486178, 514736, 525915, 669688, 686072, 652664, 691256, 712216, 783556, 796696, 863835, 901424, 980984, 1043096, 1125765
Offset: 1

Views

Author

Omar E. Pol, Jul 15 2015

Keywords

Comments

Another version of A002046.
First differs from A002046 at a(9).

Links

Crossrefs

Cf. A002046, A063990, A259180, A259933, A259953, A260086.

Formula

a(n) = A259933(2n) = A259953(n) - A259933(2n-1) = A259953(n) - A260086(n).
Showing 1-4 of 4 results.

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