A209872 Numbers whose Schwarzian arithmetic derivative is an integer.

12, 39, 55, 81, 515, 707, 1067, 1255, 1454, 1691, 1724, 2291, 2627, 2747, 2867, 3408, 4063, 5359, 6583, 7996, 8615, 9375, 11623, 11637, 12047, 12279, 13248, 14359, 14863, 15943, 17136, 20455, 23004, 27644, 32471, 37491, 39424, 49271, 52607, 53973, 53996, 54656

Offset

1,1

Comments

The sequence lists the numbers n for which the expression (n''/n')' - (1/2)*(n''/n')^2 or n'''/n' - (3/2)*(n''/n')^2 gives an integer less than zero, where n', n'', n''' are the first, second and third arithmetic derivatives.

Curiously the integer values of the Schwarzian derivative, tested up to 30 million, seem to be essentially -1, -3, -4, -13, plus sporadic occurrences of -20 (for 1113823, 2211815, 5824783, 7392799, 10057552, 11698903, 14929895, 17556823, 18135407, 23009599, 25342183), -25 (for 10350000, 12274343, 12857807, 13149527, 13387500, 13732751, 13829927, 14315687, 16159751, 17226047, 18194567, 19549151, 20419127, 20515751, 23314367, 23892551, 24470447, 26204063, 26298551, 27355607, 27530519, 29754407), -36 (for 10223447, 16286940), -43 (for 2191040, 3145719, 5242855, 14789520, 17825503) and -56 (for 1835008, 12386304).

Links

Paolo P. Lava, Table of n, a(n) for n = 1..250

H. V. Ovsienko and S. Tabachnikov, What is the Schwarzian Derivative?, AMS Notices 56 (01), 34-36.

Example

To compute the Schwarzian derivative of 1724:
1724'=1728; 1728'=6912; 6912'=34560. (6912/1728)' - (1/2)*(6912/1728)^2 = 4' - (1/2)*16 = 4 - 8 = -4 or 34560/1728 - (3/2)*16 = 20 - 3*8 = 20 - 24 = -4.

Maple

with(numtheory);
A209872:= proc(i)
local a, b, c, d, n, p, pfs;
for n from 2 to i do
  pfs:=ifactors(n)[2]; a:=n*add(op(2, p)/op(1, p), p=pfs);
  pfs:=ifactors(a)[2]; b:=a*add(op(2, p)/op(1, p), p=pfs);
  pfs:=ifactors(b)[2]; c:=b*add(op(2, p)/op(1, p), p=pfs);
  d:=c/a-3/2*(b/a)^2; if d=trunc(d) and d<>0 then lprint(n, d); fi;
od; end:
A209872(10000000);

Crossrefs

Cf. A003415, A094901, A094902, A094903, A145900.

Sequence in context: A259517 A303619 A167712 * A186779 A154266 A236267

Adjacent sequences: A209869 A209870 A209871 * A209873 A209874 A209875

Keyword

nonn

Author

Paolo P. Lava, Mar 23 2012

Status

approved