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A322256 Numbers k such that t(k) = t(k+1) where t(k) = tau(k) + sigma(k) = A007503(k) is the number of subgroups of the dihedral group of order 2k.

%I A322256 #19 Jul 25 2025 06:12:03
%S A322256 14,1334,1634,2685,33998,42818,64665,84134,109214,122073,166934,
%T A322256 289454,383594,440013,544334,605985,649154,655005,792855,845126,
%U A322256 1642154,2284814,2305557,2913105,3571905,3682622,4701537,5181045,6431732,6444873,6771405,10074477
%N A322256 Numbers k such that t(k) = t(k+1) where t(k) = tau(k) + sigma(k) = A007503(k) is the number of subgroups of the dihedral group of order 2k.
%C A322256 Jensen and Keane asked if this sequence is infinite. Jensen and Bussian suggested the calculation of this sequence as a part of a student research project.
%C A322256 Supersequence of A054004. Terms that are not in it are 845126, 14392646, 10461888478, ...
%H A322256 Amiram Eldar, <a href="/A322256/b322256.txt">Table of n, a(n) for n = 1..400</a>
%H A322256 David W. Jensen and Michael K. Keane, <a href="https://apps.dtic.mil/sti/tr/pdf/ADA222857.pdf">A Number-Theoretic Approach to Subgroups of Dihedral Groups</a>, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.
%H A322256 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.
%t A322256 t[n_] := DivisorSigma[0, n] + DivisorSigma[1, n]; tQ[n_] := t[n] == t[n + 1]; Select[Range[1000000], tQ]
%o A322256 (PARI) isok(n) = (numdiv(n)+sigma(n)) == (numdiv(n+1)+sigma(n+1)); \\ _Michel Marcus_, Dec 04 2018
%o A322256 (Magma) [n: n in [1..2*10^6] | (NumberOfDivisors(n) + SumOfDivisors(n)) eq (NumberOfDivisors(n+1) + SumOfDivisors(n+1))]; // _Vincenzo Librandi_, Dec 08 2018
%Y A322256 Cf. A007503, A054004, A083874.
%K A322256 nonn
%O A322256 1,1
%A A322256 _Amiram Eldar_, Dec 01 2018