cp's OEIS Frontend

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).

Another version of A002046.

First differs from A002046 at a(9).

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.

Also the common value of sigma(x) = sigma(y) of the amicable pairs (x < y) ordered by nondecreasing sum (x + y). See A259933.

Duplicates occur, e.g., a(32) = a(33) = 1296000.

Another version of A180164.

First differs from both A161005 and A180164 at a(9).

Each term represents the length of an interval (x,y), where x (A260086) and y (A260087) form a pair of amicable numbers (A259933). The midpoint and radius of each interval can be found in A275316 and A275470, respectively.

Each term will be even as long as there does not exist an amicable pair where x and y have opposite parity.

This sequence is a rearrangement of A066539 (which is based on A002025, A002046, and A259180). The first ten indices for which a(n) does not equal A066539(n) are n = 9, 10, 11, 15, 16, 33, 34, 35, 41, 42.

Each term represents the radius of an interval (x,y), where x (A260086) and y (A260087) form a pair of amicable numbers (A259933). The midpoint and length of each interval can be found in A275316 and A275469, respectively.

A term will be odd if and only if y-x = 2 mod 4. This occurs when x and y have the same parity but their average has the opposite parity.

This sequence is a rearrangement of A162884 (which is based on A002025, A002046, and A066539). The first ten indices for which a(n) does not equal A162884(n) are n = 9, 10, 11, 15, 16, 33, 34, 35, 41, 42.

The terms represent differences between consecutive amicable pair averages given in A275316.

Interestingly, the first two odd abundant numbers begin this sequence: a(1) = 945 = A005231(1) and a(2) = 1575 = A005231(2).

Of the first 141 terms, 4 are odd, 137 are even, 136 are abundant, 2 are deficient [specifically, a(27) = 31203 and a(28) = 31293], 7 numbers occur twice [specifically, a(53) = a(59) = 1728, a(25) = a(136) = 3024, a(21) = a(127) = 5184, a(31) = a(137) = 12960, a(34) = a(41) = 16200, a(74) = a(77) = 20736, and a(66) = a(104) = 156240], 3 numbers occur three times [specifically, a(32) = a(107) = a(139) = 0, a(35) = a(42) = a(52) = 51840, and a(30) = a(86) = a(97) = 110880], and every number is divisible by 3.

a(n) = A275066(n) for 41 of the first 141 indices: n = 1, 2, 3, 4, 5, 6, 7, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 36, 37, 38, 39, 43, 44, 48, 49, 50, 57, 58, 59, 64, 65, 95, 120, 121.

a(n) = -A275066(n) for 9 of the first 141 indices: n = 15, 41, 46, 55, 67, 70, 81, 86, 141.