A259933 -id:A259933 - cp's OEIS frontend

A262624 Even amicable numbers.

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 17296, 18416, 63020, 66928, 66992, 76084, 79750, 88730, 122368, 123152, 141664, 142310, 153176, 168730, 171856, 176272, 176336, 180848, 185368, 196724, 202444, 203432, 280540, 308620, 319550, 356408, 365084, 389924, 399592, 430402, 437456, 455344

Offset: 1

Omar E. Pol, Oct 02 2015

nonn

Even numbers that are also amicable numbers.
Intersection of A005843 and A063990.
The first time a pair (x, y) of even amicable numbers ordered by its first element is not adjacent is x = 63020, y = 76084 which correspond to a(15) and a(18), respectively.

Cf. A002025, A002046, A005843, A063990, A259180, A259933, A262625.

t(n)=sigma(n)-n;
is(n)={local(a); a=t(n); a<>n && t(a)==n};
for(n=1, 1e6, if( n%2 == 0 && is(n), print1(n", "))) \\ Altug Alkan, Oct 16 2015

A275472 First differences of A275316.

Original entry on oeis.org

945, 1575, 2520, 1008, 4500, 2640, 4416, 49104, 2160, 432, 9072, 5616, 28080, 10440, 8280, 16380, 8100, 18576, 4464, 15840, 5184, 123228, 26460, 25704, 3024, 68400, 31203, 31293, 15264, 110880, 12960, 0, 9720, 16200, 51840, 35640, 71880, 57288, 49032, 81000

Offset: 1

Timothy L. Tiffin, Jul 29 2016

nonn

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.

			a(9) = A275316(10) - A275316(9) = 69120 - 66960 = 2160.

Timothy L. Tiffin, Table of n, a(n) for n = 1..141
VaxaSoftware, List of amicable numbers from 1 to 20,000,000 [142 pairs].

Cf. A005231, A260086, A260087, A259933, A275066, A275316.

a(n) = A275316(n+1) - A275316(n).

A273259 Twin amicable numbers (m,n) where there is not any part of another amicable pair between m and n.

Original entry on oeis.org

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 66928, 66992, 122368, 123152, 196724, 202444, 437456, 455344, 469028, 486178, 503056, 514736, 522405, 525915, 643336, 652664, 802725, 863835, 998104, 1043096, 1077890, 1099390

Offset: 1

Zoltan Galantai, May 18 2016

nonn

The sequence lists the twin amicable number pairs in increasing order. So opposite to the list of the amicable numbers, the pairs (m, n) are necessarily adjacent to each other in this list.

First differs from A259933 at a(19). - Omar E. Pol, May 19 2016

			The (63020, 76084) amicable pair does not qualify since it overlaps with the (66928, 66992) pair.
On the other hand, 66928 and 66992 are contiguous, thus the (66928, 66992) amicable pair belongs to the sequence.

Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc, 1996, pp. 113-121.

Zoltan Galantai, Table of n, a(n) for n = 1..150
Bela Galantai, Software source code written in Clipper
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A002025 (smaller), A002046 (and larger of an amicable pair).

Cf. A063990 (amicable numbers), A259180 (amicable pairs), A259933.

A359334 Amicable numbers k that can be expressed as a sum k = x+y = A001065(x) + A001065(y) and a sum k = z+t = A001065(z) + A001065(t) where (x, y, z, t) are parts of two amicable pairs and A001065(i) is the sum of the aliquot parts of i.

Original entry on oeis.org

67212, 1296000, 20528640, 37739520, 75479040, 321408000, 348364800, 556839360, 579156480, 638668800, 661893120, 761177088, 796340160, 883872000, 1181174400, 1282417920, 2068416000, 2395008000, 2682408960, 3155023872, 3599769600, 4049740800, 4606156800, 4716601344

Offset: 1

Zoltan Galantai, Dec 26 2022

nonn

From Michel Marcus, Dec 31 2022: (Start)
In other words, numbers k that can be expressed as a sum k = x+y = z+t where either (x,y) and (z,t), or (x,z) and (y,t), are 2 amicable pairs.
Note that there is currently a single instance of the case (x,z) and (y,t), and this corresponds to the value 64 that appears twice in A066539.
The other terms correspond to values appearing at least twice in A180164.
There are instances where it can appear 3 times, and the least instance is 64795852800 for the 3 amicable pairs [29912035725, 34883817075], [31695652275, 33100200525], [32129958525, 32665894275].
There are instances where it can appear 6 times, and the least instance is 4169926656000 for the 6 amicable pairs [1953433861918, 2216492794082], [1968039941816, 2201886714184], [1981957651366, 2187969004634], [1993501042130, 2176425613870], [2046897812505, 2123028843495], [2068113162038, 2101813493962]. (End)

			67212 is a term because 67212 = 220 + 66992 = 284 + 66928 where (220, 284) and (66928, 66992) are two amicable pairs.
1296000 is a term because 1296000 = 609928 + 686072 = 643336 + 652664 where (609928, 686072) and (643336, 652664) are two amicable pairs.

Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc, 1996, pp. 113-121.

Michel Marcus, Table of n, a(n) for n = 1..233
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A002025, A063990, A259180, A259933, A036471, A180164, A001065, A066539.

More terms from Amiram Eldar, Dec 31 2022

cp's OEIS Frontend

A262624 Even amicable numbers.

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A275472 First differences of A275316.

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A273259 Twin amicable numbers (m,n) where there is not any part of another amicable pair between m and n.

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A359334 Amicable numbers k that can be expressed as a sum k = x+y = A001065(x) + A001065(y) and a sum k = z+t = A001065(z) + A001065(t) where (x, y, z, t) are parts of two amicable pairs and A001065(i) is the sum of the aliquot parts of i.

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