cp's OEIS Frontend

A188145 Solutions of the equation n" - n' - n = 0, where n' and n" are the first and second arithmetic derivatives (see A003415).

%I A188145 #19 Jun 04 2016 06:55:54
%S A188145 0,20,135,164,1107,15625,43692,128125,188228,294921,1270539,4117715,
%T A188145 33765263,34134375,147053125,8995560189,19348535652,38753462951
%N A188145 Solutions of the equation n" - n' - n = 0, where n' and n" are the first and second arithmetic derivatives (see A003415).
%C A188145 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.
%C A188145 a(16) > 2*10^9. - _Donovan Johnson_, Apr 30 2011
%C A188145 a(19) > 10^11. - _Giovanni Resta_, Jun 04 2016
%e A188145 n=20, n’=24, n”=44 -> 44-24-20=0;  n=135, n’=162, n”=297 -> 297-162-135=0
%p A188145 readlib(ifactors):
%p A188145 Der:= proc(n)
%p A188145 local a,b,i,p,pfs;
%p A188145 for i from 0 to n do
%p A188145   if i<=1 then a:=0;
%p A188145   else pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs) ;
%p A188145   fi;
%p A188145   if a<=1 then b:=0;
%p A188145   else pfs:=ifactors(a)[2]; b:=a*add(op(2,p)/op(1,p),p=pfs) ;
%p A188145   fi;
%p A188145   if b-a=i then lprint(i,a,b); fi;
%p A188145 od
%p A188145 end:
%p A188145 Der(10000000);
%o A188145 (Haskell)
%o A188145 import Data.List (zipWith3, elemIndices)
%o A188145 a188145 n = a188145_list !! (n-1)
%o A188145 a188145_list = elemIndices 0 $ zipWith3 (\x y z -> x - y - z)
%o A188145    (map a003415 a003415_list) a003415_list [0..]
%o A188145 -- _Reinhard Zumkeller_, May 10 2011
%Y A188145 Cf. A003415, A007850, A165558, A165561, A165562, A189639, A189710.
%K A188145 nonn,more
%O A188145 1,2
%A A188145 _Paolo P. Lava_, Mar 22 2011
%E A188145 a(13)-a(15) from _Donovan Johnson_, Apr 30 2011
%E A188145 Corrected a(9) and a(16)-a(18) from _Giovanni Resta_, Jun 04 2016