A336509 Even squarefree numbers k such that d_{i+1}/d_i < 2 for all 1 < i < tau(k) - 1, where 1 = d_1 < d_2 < ... < d_tau(k) = k are the divisors of k, and tau(k) is their number (A000005).

6, 30, 210, 330, 390, 510, 570, 690, 870, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 8610, 8970, 9030, 9570, 9690, 9870, 10230, 11130, 11310, 11730, 12090, 12210, 12390, 12810, 13110, 13530, 14070, 14190, 14430, 14790, 14910

Offset: 1

Author

Amiram Eldar, Jul 23 2020

Keywords

• nonn

Comments

Thompson (2012) called these numbers "strictly 2-dense numbers" and proved that they are all phi-practical numbers (A260653).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Lola Thompson, Polynomials with divisors of every degree, Journal of Number Theory, Vol. 132, No. 5 (2012), pp. 1038-1053.
• Lola Thompson, Variations on a question concerning the degrees of divisors of x^n - 1, Journal de Théorie des Nombres de Bordeaux, Vol. 26, No. 1 (2014), pp. 253-267.

Crossrefs

Subsequence of A174973 and A260653.

Cf. A000005, A005117, A039956.

Programs

Mathematica

sdQ[n_] := SquareFreeQ[n] && Length[(d = Rest @ Most @ Divisors[n])] >0 && Max[Rest[d]/Most[d]] < 2; Select[Range[2, 15000, 2], sdQ]