cp's OEIS Frontend

A274063 Numbers whose periodic derivative is equal to the arithmetic derivative.

%I A274063 #13 Jun 21 2016 16:03:21
%S A274063 0,1,25,26,51,119,218,771,1754,1799,1921,7967,16147,32639,128129,
%T A274063 196611,458759,1044143,2031647,7190234,8323199,33464867,536581571,
%U A274063 536813567,1073691551,2145328183,7202169026,8746826298,17179612627,68719005499,797299610790
%N A274063 Numbers whose periodic derivative is equal to the arithmetic derivative.
%C A274063 Solution of the equation A003415(n) = A038556(n).
%e A274063 25 in base 2 is 11001 and its periodic derivative is (1+1)(1+0)(0+0)(0+1)(1+1) -> 01010 that is 10 in base 10 and 10 is also the arithmetic derivative of 25.
%p A274063 with(numtheory): P:=proc(q) local a,b,i,n,p;
%p A274063 for n from 0 to q do a:=0; b:=convert(n,base,2); b:=[1,op(b)];
%p A274063 for i to nops(b)-1 do a:=a+((b[i]+b[i+1]) mod 2)*2^(i-1); od;
%p A274063 if a=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]) then print(n); fi;
%p A274063 od; end: P(10^6);
%t A274063 Select[Range[0, 10^6], Function[n, If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]] == FromDigits[Thread[BitXor[#, RotateLeft@ #]], 2] &@ IntegerDigits[n, 2]]] (* _Michael De Vlieger_, Jun 10 2016 after _Michael Somos_ at A003415 and _Jean-François Alcover_ at A038556 *)
%Y A274063 Cf. A003415, A038556, A273993.
%K A274063 nonn
%O A274063 1,3
%A A274063 _Paolo P. Lava_, Jun 09 2016
%E A274063 a(23)-a(31) from _Giovanni Resta_, Jun 19 2016