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.

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

Author

Zoltan Galantai, Dec 26 2022

Keywords

• nonn

Comments

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)

Examples

			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.
		

References

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

Links

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

Crossrefs

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

Extensions

More terms from Amiram Eldar, Dec 31 2022