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

1,3

Comments

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

Links

Table of n, a(n) for n=1..31.

Example

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.

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 *)

Crossrefs

Cf. A003415, A038556, A273993.

Sequence in context: A042260 A042262 A042264 * A042256 A042258 A042254

Adjacent sequences: A274060 A274061 A274062 * A274064 A274065 A274066

Keyword

nonn

Author

Paolo P. Lava, Jun 09 2016

Extensions

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

Status

approved