A322256 - OEIS

%I #12 Sep 08 2022 08:46:23

%S 14,1334,1634,2685,33998,42818,64665,84134,109214,122073,166934,

%T 289454,383594,440013,544334,605985,649154,655005,792855,845126,

%U 1642154,2284814,2305557,2913105,3571905,3682622,4701537,5181045,6431732,6444873,6771405,10074477

%N Number such that t(n) = t(n+1) where t(n) = tau(n) + sigma(n) = A007503(n) is the number of subgroups of the dihedral group of order 2n.

%C 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 Supersequence of A054004. Terms that are not in it are 845126, 14392646, 10461888478, ...

%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.

%t t[n_] := DivisorSigma[0, n] + DivisorSigma[1, n]; tQ[n_] := t[n] == t[n + 1]; Select[Range[1000000], tQ]

%o (PARI) isok(n) = (numdiv(n)+sigma(n)) == (numdiv(n+1)+sigma(n+1)); \\ _Michel Marcus_, Dec 04 2018

%o (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 Cf. A007503, A054004, A083874.

%K nonn

%O 1,1

%A _Amiram Eldar_, Dec 01 2018