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

1,2

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

Links

Table of n, a(n) for n=1..18.

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

Crossrefs

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

Sequence in context: A374709 A219574 A356272 * A168178 A085284 A105573

Adjacent sequences: A188142 A188143 A188144 * A188146 A188147 A188148

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