login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Solutions of the equation n'' = tau(n) * n', where n' and n'' are the first and the second arithmetic derivative of n.
0

%I #4 Oct 27 2013 02:34:33

%S 1,1255,4063,5359,6583,8615,11623,12047,14359,14863,15943,27644,32471,

%T 49271,52607,81599,98471,101687,117647,164327,173447,176471,203327,

%U 209207,235271,246647,277271,301607,343271,355871,358367,360623,378047,392471,401927,406607

%N Solutions of the equation n'' = tau(n) * n', where n' and n'' are the first and the second arithmetic derivative of n.

%e For n = 6583 we have tau(n) = 4, n’ = 256, n’’= 1024 and 1024 = 4 * 256.

%p with(numtheory); P:= proc(q) local a1, a2, n, p;

%p for n from 1 to q do a1:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); ;

%p a2:=a1*add(op(2,p)/op(1,p),p=ifactors(a1)[2]);

%p if a2=tau(n)*a1 then print(n); fi; od; end: P(10^6);

%Y Cf. A000005, A003415, A068346.

%K nonn

%O 1,2

%A _Paolo P. Lava_, Oct 25 2013