A060798 Numbers k such that difference between the upper and lower central divisors of k is 1.

2, 4, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450

Offset

1,1

Comments

From David A. Corneth, Sep 02 2018: (Start)

Theorem: a(1) = 2, a(2) = 4; a(n) = n*(n-1) for n > 2.

Proof:

If a(n) is a square m^2 then the upper central divisor is m and by definition of the sequence the lower one is m-1. But m-1 and m are coprime, and (m-1)|m^2 implies m-1 = 1, i.e. a(n) = 4.

if a(n) is not a square then it has an even number of divisors with m and m-1 the central divisors, so it has the form m*(m-1), i.e. is oblong (see A002378). QED (End)

Links

Harvey P. Dale, Table of n, a(n) for n = 1..2000 [First 1000 terms from Harry J. Smith]

Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Formula

Solutions to A033677(k) - A060775(k) = 1, where k = j*(j+1) and at least one of j and j+1 is composite.

Except at n < 5, this sequence satisfies a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2). - Georgi Guninski, Jun 07 2010 [This follows from Corneth's theorem above. - N. J. A. Sloane, Sep 02 2018]

G.f.: (2*x^2-2*x+1)*(x^3-x^2-x-1) / (x-1)^3. - Colin Barker, Apr 16 2014 [This follows from Corneth's theorem above. - N. J. A. Sloane, Sep 02 2018]

Example

The divisors of 2 are 1 and 2, so the upper central divisor is 2 and the lower central divisor is 1, so a(1)=2 is a member.
k = 4032 = 2*2*2*2*2*2*3*3*7 is here because its central divisors (the 21st and 22nd divisors) are {63,64} which differ by 1.

Mathematica

dulcdQ[n_]:=Module[{d=Divisors[n], len}, len=Floor[Length[d]/2]; d[[len+1]] - d[[len]]==1]; Select[Range[2500], dulcdQ] (* or *) Join[{2, 4}, Table[ n(n-1), {n, 3, 60}]] (* after David A. Corneth's comment and formula *) (* Harvey P. Dale, Aug 28 2018 *)

Prog

(PARI) { n=-1; for (m=1, 999000, d=divisors(m); if (m==1 || (d[1 + length(d)\2] - d[length(d)\2]) == 1, write("b060798.txt", n++, " ", m)); ) } \\ Harry J. Smith, Jul 13 2009
(PARI) first(n) = res = List([2, 4]); for(i = 3, n, listput(res, i*(i-1))); res \\ David A. Corneth, Sep 02 2018

Crossrefs

Cf. A000196, A002378, A033677, A060775.

Sequence in context: A307067 A358101 A358100 * A134320 A353561 A329227

Adjacent sequences: A060795 A060796 A060797 * A060799 A060800 A060801

Keyword

nonn,easy

Author

Labos Elemer, Apr 27 2001

Extensions

Start and offset changed by N. J. A. Sloane, Sep 02 2018 at the suggestion of Harvey P. Dale. Further edited by N. J. A. Sloane, Sep 02 2018

Status

approved