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A274063 Numbers whose periodic derivative is equal to the arithmetic derivative. 0

%I

%S 0,1,25,26,51,119,218,771,1754,1799,1921,7967,16147,32639,128129,

%T 196611,458759,1044143,2031647,7190234,8323199,33464867,536581571,

%U 536813567,1073691551,2145328183,7202169026,8746826298,17179612627,68719005499,797299610790

%N Numbers whose periodic derivative is equal to the arithmetic derivative.

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

%e 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 with(numtheory): P:=proc(q) local a,b,i,n,p;

%p for n from 0 to q do a:=0; b:=convert(n,base,2); b:=[1,op(b)];

%p for i to nops(b)-1 do a:=a+((b[i]+b[i+1]) mod 2)*2^(i-1); od;

%p if a=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]) then print(n); fi;

%p od; end: P(10^6);

%t 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-Fran├žois Alcover_ at A038556 *)

%Y Cf. A003415, A038556, A273993.

%K nonn

%O 1,3

%A _Paolo P. Lava_, Jun 09 2016

%E a(23)-a(31) from _Giovanni Resta_, Jun 19 2016

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Last modified October 1 00:54 EDT 2020. Contains 337440 sequences. (Running on oeis4.)