A013654 Positive nonsquare integers k such that each term of the regular continued fraction of sqrt(k) divides k.

2, 8, 12, 24, 30, 48, 56, 80, 90, 120, 132, 168, 182, 224, 240, 288, 306, 360, 380, 440, 462, 528, 552, 624, 650, 728, 756, 840, 870, 960, 992, 1088, 1122, 1224, 1260, 1368, 1406, 1520, 1560, 1680, 1722, 1848, 1892, 2024, 2070, 2208, 2256, 2400, 2450, 2600

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

a(n) = n*(n+1) if n is odd and n*(n+2) if n is even. - Wesley Ivan Hurt, Nov 19 2013

From Rolf Knobel, Dec 27 2023: (Start)

The first term of the continued fraction also divides k since it is half the last term of the periodic part. Except for a(1), the period is 2 (see A013642). -

Interleaving of A033996, A002939. Alternating mixture of A005563, A002378. (End)

References

• H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th edition, 1999, Table 1.
• Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A002378, A002939, A005563, A013642, A033996.

Programs

Maple

A013654:=n->n*(n+1+(1+(-1)^n)/2); seq(A013654(n), n=1..100); # Wesley Ivan Hurt, Nov 19 2013

Mathematica

Table[n(n+1+(1+(-1)^n)/2), {n,100}] (* Wesley Ivan Hurt, Nov 19 2013 *)
Table[If[OddQ[n],n(n+1),n(n+2)],{n,50}] (* Harvey P. Dale, May 15 2021 *)

Formula

a(2n) = 2*(n+1)*(2n), a(2n+1) = 2*(n+1)*(2n+1). - Frank Ellermann, Feb 22 2002

G.f.: (-2-6*x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009

a(n) = n * (n + 1 + (1+(-1)^n)/2). - Wesley Ivan Hurt, Nov 19 2013

Extensions

More terms from David W. Wilson

Name clarified by Rolf Knobel, Dec 27 2023