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

0, 20, 135, 164, 1107, 15625, 43692, 128125, 188228, 294921, 1270539, 4117715, 33765263, 34134375, 147053125, 8995560189, 19348535652, 38753462951

Offset: 1

Author

Paolo P. Lava, Mar 22 2011

Keywords

• nonn
• more

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

Examples

			n=20, n’=24, n”=44 -> 44-24-20=0;  n=135, n’=162, n”=297 -> 297-162-135=0
		

Crossrefs

Cf. A003415, A007850, A165558, A165561, A165562, A189639, A189710.

Programs

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

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);

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