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
Keywords
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.
Programs
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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);
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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
Comments