A246769 Composite x such that [(x-1)’ + (x+1)’] / x’ is an integer, where x’ is the arithmetic derivative of x.

10, 14, 1234, 9302, 15621, 45069, 275825, 1496771, 3901747, 3965572, 4800842, 12089923, 13725353, 60247178, 86123531, 141164047, 400351433, 577144967, 733863869, 797811821, 1107698663, 1230427279, 1745874461, 1963869823, 2069222929, 2568664561, 3288702721

Offset

1,1

Comments

The ratio for the first numbers is 1, 1, 2, 1, 5, 5, 5, 47, 39, 1, 2, 63, 43

a(202) > 5*10^13. - Hiroaki Yamanouchi, Aug 27 2015

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..201

Example

The arithmetic derivatives of 1233, 1234, 1235 are 831, 619, 407 and (831 + 407) / 619 = 2.

Maple

for x from 2 do
    if not isprime(x) then
        if modp(A003415(x-1)+A003415(x+1), A003415(x)) = 0 then
            print(x) ;
        end if;
    end if;
end do: # R. J. Mathar, Nov 11 2014

Mathematica

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 *)

Prog

(PARI) der(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);
lista(nn) = forcomposite(n=2, nn, if (! ((der(n-1) + der(n+1)) % der(n)), print1(n, ", "))); \\ Michel Marcus, Aug 27 2015

Crossrefs

Cf. A003415.

Sequence in context: A219682 A167334 A236145 * A322566 A004475 A216118

Adjacent sequences: A246766 A246767 A246768 * A246770 A246771 A246772

Keyword

nonn

Author

Paolo P. Lava, Sep 03 2014

Extensions

Definition corrected by R. J. Mathar, Nov 11 2014

a(14)-a(27) from Hiroaki Yamanouchi, Aug 27 2015

Status

approved