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Lesser number of dihedral amicable pairs: numbers (m, n) such that t(m) = t(n) = m + n, where t(n) = sigma(n) + d(n).
1

%I #29 Dec 10 2018 09:00:50

%S 144,300,10434,15774,17034,21032,22088,35394,36872,65324,67628,153868,

%T 188468,254526,379026,483812,492414,905212,1090528,1198180,1514212,

%U 1634262,1886046,1898420,2013414,2184860,2191588,2316546,2596448,2816156,3340024,3854886

%N Lesser number of dihedral amicable pairs: numbers (m, n) such that t(m) = t(n) = m + n, where t(n) = sigma(n) + d(n).

%C Jensen and Bussian suggested the calculation of this sequence as a part of a student research project.

%H David W. Jensen and Michael K. Keane, <a href="http://www.dtic.mil/docs/citations/ADA222857">A Number-Theoretic Approach to Subgroups of Dihedral Groups</a>, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.

%H David W. Jensen and Eric R. Bussian, <a href="http://www.jstor.org/stable/2686678">A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups</a>, The College Mathematics Journal, Vol. 23, No. 2 (1992), pp. 150-152.

%e 144 is in the sequence since (144, 274) is a pair of dihedral amicable numbers: sigma(144) + d(144) = 403 + 15 = 418, sigma(274) + d(274) = 414 + 4 = 418, and 144 + 274 = 418.

%t t[n_] := DivisorSigma[0,n] + DivisorSigma[1,n] - n; s={}; Do[n = t[m]; If[n>m && t[n]==m, AppendTo[s,m]], {m, 1, 100000}];s

%o (PARI) f(n) = numdiv(n) + sigma(n) - n;

%o isok(n) = my(nn = f(n)); (nn > n) && (n == f(nn)); \\ _Michel Marcus_, Dec 04 2018

%Y Cf. A007503, A083874, A322254.

%K nonn

%O 1,1

%A _Amiram Eldar_, Dec 01 2018