A189639 -id:A189639 - cp's OEIS frontend

A189710 Numbers n such that n'' = n'-1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 185, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Giorgio Balzarotti, Apr 25 2011

nonn

The composite numbers in the sequences are: 9, 185, 341, 377, 437, 9005, 30413, 33953, 41009, 51533, 82673, 92909,....

			9' = 6, 9''= 6'= 5, 9" = 9'- 1 -> 9 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003415, A007850, A054377, A189639.

import Data.List (elemIndices)
a189710 n = a189710_list !! (n-1)
a189710_list = elemIndices 0 $
   zipWith (-) (map a003415 a003415_list) (map pred a003415_list)
-- Reinhard Zumkeller, May 09 2011

#using Michael B. Porter's code from A003415
der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2])
for i from 1 to n do a:=der(der(i))-der(i)+1; if a=0 then j:=j+1; B[j]:=i; end if od

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

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Mar 22 2011

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

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

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

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

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);

a(13)-a(15) from Donovan Johnson, Apr 30 2011

Corrected a(9) and a(16)-a(18) from Giovanni Resta, Jun 04 2016

cp's OEIS Frontend

A189710 Numbers n such that n'' = n'-1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

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