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

0, 1, 25, 26, 51, 119, 218, 771, 1754, 1799, 1921, 7967, 16147, 32639, 128129, 196611, 458759, 1044143, 2031647, 7190234, 8323199, 33464867, 536581571, 536813567, 1073691551, 2145328183, 7202169026, 8746826298, 17179612627, 68719005499, 797299610790

Offset: 1

Author

Paolo P. Lava, Jun 09 2016

Keywords

• nonn

Comments

Solution of the equation A003415(n) = A038556(n).

Examples

			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.
		

Crossrefs

Cf. A003415, A038556, A273993.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,i,n,p;
for n from 0 to q do a:=0; b:=convert(n,base,2); b:=[1,op(b)];
for i to nops(b)-1 do a:=a+((b[i]+b[i+1]) mod 2)*2^(i-1); od;
if a=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]) then print(n); fi;
od; end: P(10^6);

Mathematica

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 *)

Extensions

a(23)-a(31) from Giovanni Resta, Jun 19 2016