A212127 - OEIS

%I #36 Feb 10 2014 02:09:05

%S 1,2,3,5,7,11,12,13,17,18,19,23,29,31,37,41,43,47,53,59,61,67,71,73,

%T 79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,

%U 173,179,181,191,193,197,199,211,220,223,227,229,233,239,241,251,257

%N Numbers n whose arithmetic derivative equals the sum of its proper divisors.

%C Numbers n such that A003415(n) = A001065(n). Also numbers n such that A211991(n) = 0. By definition, all prime numbers are in the sequence. Nonprime numbers in the sequence are 1, 12, 18, 220,...

%H Paolo P. Lava, <a href="/A212127/b212127.txt">Table of n, a(n) for n = 1..1000</a>

%e The arithmetic derivative of 12 is equal to 16 (see A003415). On the other hand the sum of proper divisors of 12 is equal to 16 since 1+2+3+4+6 = 16, so 12 is in the sequence.

%p with(numtheory);

%p A212127:=proc(i)

%p local n, p;

%p for n from 1 to i do

%p   if sigma(n)/n-1=add(op(2,p)/op(1,p),p=ifactors(n)[2]) then print(n);

%p fi; od; end:

%p A212127(1000);  # _Paolo P. Lava_, Jan 04 2012

%t dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Select[Range[300], dn[#] == DivisorSigma[1, #] - # &] (* _T. D. Noe_, Dec 27 2012 *)

%Y Cf. A000040, A000203, A001065, A003415, A211991.

%K nonn

%O 1,2

%A _Omar E. Pol_, Dec 18 2012