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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.

%I A180330 #15 Jul 26 2025 08:04:02
%S A180330 2620,10744,66928,2082464,7677248,1750776704,749380864,7074650624,
%T A180330 25937232896,161899964416,3949032574976,56691934109696,
%U A180330 162222327218176,5469697508737024,21547979005558784,48336727662002176,2961911925308653568,5591728346540539904
%N 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.
%C A180330 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?
%H A180330 Sergei Chernykh, <a href="https://sech.me/ap/">Amicable numbers list</a>.
%H A180330 Jan Munch Pedersen, <a href="http://62.198.248.44/aliquot/apstat/apreg22.txt">Regular type (2,2) amicable pairs</a>.
%H A180330 Herman J. J. te Riele, <a href="http://www.ams.org/journals/mcom/1984-42-165/S0025-5718-1984-0725997-0/">On generating new amicable pairs from given amicable pairs</a>, Math. Comp. 42 (1984), 219-223.
%Y A180330 Cf. A002025, A063990, A180331.
%K A180330 nonn
%O A180330 2,1
%A A180330 _T. D. Noe_, Sep 07 2010
%E A180330 a(18)-a(19) from Chernykh's database added by _Amiram Eldar_, Jul 26 2025