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A188145
Solutions of the equation n" - n' - n = 0, where n' and n" are the first and second arithmetic derivatives (see A003415).
1
0, 20, 135, 164, 1107, 15625, 43692, 128125, 188228, 294921, 1270539, 4117715, 33765263, 34134375, 147053125, 8995560189, 19348535652, 38753462951
OFFSET
1,2
COMMENTS
Solutions of the similar equation n”-n’+n=0 are 30, 858, 1722, etc., apparently Giuga numbers whose derivative is a prime number. In fact the equation can be rewritten as n'=n+n" and if n"=1 it is the conjecture in A007850.
a(16) > 2*10^9. - Donovan Johnson, Apr 30 2011
a(19) > 10^11. - Giovanni Resta, Jun 04 2016
EXAMPLE
n=20, n’=24, n”=44 -> 44-24-20=0; n=135, n’=162, n”=297 -> 297-162-135=0
MAPLE
readlib(ifactors):
Der:= proc(n)
local a, b, i, p, pfs;
for i from 0 to n do
if i<=1 then a:=0;
else pfs:=ifactors(i)[2]; a:=i*add(op(2, p)/op(1, p), p=pfs) ;
fi;
if a<=1 then b:=0;
else pfs:=ifactors(a)[2]; b:=a*add(op(2, p)/op(1, p), p=pfs) ;
fi;
if b-a=i then lprint(i, a, b); fi;
od
end:
Der(10000000);
PROG
(Haskell)
import Data.List (zipWith3, elemIndices)
a188145 n = a188145_list !! (n-1)
a188145_list = elemIndices 0 $ zipWith3 (\x y z -> x - y - z)
(map a003415 a003415_list) a003415_list [0..]
-- Reinhard Zumkeller, May 10 2011
KEYWORD
nonn,more
AUTHOR
Paolo P. Lava, Mar 22 2011
EXTENSIONS
a(13)-a(15) from Donovan Johnson, Apr 30 2011
Corrected a(9) and a(16)-a(18) from Giovanni Resta, Jun 04 2016
STATUS
approved