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.

14, 1334, 1634, 2685, 33998, 42818, 64665, 84134, 109214, 122073, 166934, 289454, 383594, 440013, 544334, 605985, 649154, 655005, 792855, 845126, 1642154, 2284814, 2305557, 2913105, 3571905, 3682622, 4701537, 5181045, 6431732, 6444873, 6771405, 10074477

Offset: 1

Author

Amiram Eldar, Dec 01 2018

Keywords

• nonn

Comments

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.

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..400
• David W. Jensen and Michael K. Keane, A Number-Theoretic Approach to Subgroups of Dihedral Groups, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.
• David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, The College Mathematics Journal, Vol. 23, No. 2 (1992), pp. 150-152.

Crossrefs

Cf. A007503, A054004, A083874.

Programs

Magma

[n: n in [1..2*10^6] | (NumberOfDivisors(n) + SumOfDivisors(n)) eq (NumberOfDivisors(n+1) + SumOfDivisors(n+1))]; // Vincenzo Librandi, Dec 08 2018

Mathematica

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

PARI

isok(n) = (numdiv(n)+sigma(n)) == (numdiv(n+1)+sigma(n+1)); \\ Michel Marcus, Dec 04 2018