%I #24 Aug 29 2015 01:02:42
%S 10,14,1234,9302,15621,45069,275825,1496771,3901747,3965572,4800842,
%T 12089923,13725353,60247178,86123531,141164047,400351433,577144967,
%U 733863869,797811821,1107698663,1230427279,1745874461,1963869823,2069222929,2568664561,3288702721
%N Composite x such that [(x-1)’ + (x+1)’] / x’ is an integer, where x’ is the arithmetic derivative of x.
%C The ratio for the first numbers is 1, 1, 2, 1, 5, 5, 5, 47, 39, 1, 2, 63, 43
%C a(202) > 5*10^13. - _Hiroaki Yamanouchi_, Aug 27 2015
%H Hiroaki Yamanouchi, <a href="/A246769/b246769.txt">Table of n, a(n) for n = 1..201</a>
%e The arithmetic derivatives of 1233, 1234, 1235 are 831, 619, 407 and (831 + 407) / 619 = 2.
%p for x from 2 do
%p if not isprime(x) then
%p if modp(A003415(x-1)+A003415(x+1),A003415(x)) = 0 then
%p print(x) ;
%p end if;
%p end if;
%p end do: # _R. J. Mathar_, Nov 11 2014
%t f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; lim = 300000; Select[Rest@ Complement[Range@ lim, Prime@ Range@ PrimePi@ lim], IntegerQ[(f[# - 1] + f[# + 1])/f@ #] &] (* _Michael De Vlieger_, Aug 27 2015, after _Michael Somos_ at A003415 *)
%o (PARI) der(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);
%o lista(nn) = forcomposite(n=2, nn, if (! ((der(n-1) + der(n+1)) % der(n)), print1(n, ", "))); \\ _Michel Marcus_, Aug 27 2015
%Y Cf. A003415.
%K nonn
%O 1,1
%A _Paolo P. Lava_, Sep 03 2014
%E Definition corrected by _R. J. Mathar_, Nov 11 2014
%E a(14)-a(27) from _Hiroaki Yamanouchi_, Aug 27 2015