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A209872
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Numbers whose Schwarzian arithmetic derivative is an integer.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MAPLE
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with(numtheory);
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:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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