A180330 Smallest amicable number of the form 2^n * p * q for which the larger member of the amicable pair has the same form, where p and q are distinct odd primes.

2620, 10744, 66928, 2082464, 7677248, 1750776704, 749380864, 7074650624, 25937232896, 161899964416, 3949032574976, 56691934109696, 162222327218176, 5469697508737024, 21547979005558784, 48336727662002176, 2961911925308653568, 5591728346540539904

Offset: 2

Author

T. D. Noe, Sep 07 2010

Keywords

• nonn

Comments

That is, the amicable pair is (2^n pq, 2^n rs) for odd primes p, q, r, s. See A180331 for the numbers 2^n rs. It is easy to show that the four primes must satisfy the equation (p+1)(q+1) = (r+1)(s+1). These amicable pairs are a subset of the regular type (2,2) pairs, which are cataloged by Pedersen. These amicable pairs can be found by using Herman te Riele's method 2. Amicable pairs of this form are known for 1 < n < 49. Do they exist for all n?

Links

• Sergei Chernykh, Amicable numbers list.
• Jan Munch Pedersen, Regular type (2,2) amicable pairs.
• Herman J. J. te Riele, On generating new amicable pairs from given amicable pairs, Math. Comp. 42 (1984), 219-223.

Crossrefs

Cf. A002025, A063990, A180331.

Extensions

a(18)-a(19) from Chernykh's database added by Amiram Eldar, Jul 26 2025