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%I #18 Jan 21 2022 05:06:16
%S 12,39,55,81,515,707,1067,1255,1454,1691,1724,2291,2627,2747,2867,
%T 3408,4063,5359,6583,7996,8615,9375,11623,11637,12047,12279,13248,
%U 14359,14863,15943,17136,20455,23004,27644,32471,37491,39424,49271,52607,53973,53996,54656
%N Numbers whose Schwarzian arithmetic derivative is an integer.
%C 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.
%C 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).
%H Paolo P. Lava, <a href="/A209872/b209872.txt">Table of n, a(n) for n = 1..250</a>
%H H. V. Ovsienko and S. Tabachnikov, <a href="http://www.ams.org/notices/200901/tx090100034p.pdf">What is the Schwarzian Derivative?</a>, AMS Notices 56 (01), 34-36.
%e To compute the Schwarzian derivative of 1724:
%e 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.
%p with(numtheory);
%p A209872:= proc(i)
%p local a,b,c,d,n,p,pfs;
%p for n from 2 to i do
%p pfs:=ifactors(n)[2]; a:=n*add(op(2,p)/op(1,p),p=pfs);
%p pfs:=ifactors(a)[2]; b:=a*add(op(2,p)/op(1,p),p=pfs);
%p pfs:=ifactors(b)[2]; c:=b*add(op(2,p)/op(1,p),p=pfs);
%p d:=c/a-3/2*(b/a)^2; if d=trunc(d) and d<>0 then lprint(n,d); fi;
%p od; end:
%p A209872(10000000);
%Y Cf. A003415, A094901, A094902, A094903, A145900.
%K nonn
%O 1,1
%A _Paolo P. Lava_, Mar 23 2012
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