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A255683 Sum of the binary numbers whose digits are cyclic permutations of the binary expansion of n.

%I A255683 #36 Jul 08 2025 07:49:38
%S A255683 1,3,6,7,14,14,21,15,30,30,45,30,45,45,60,31,62,62,93,62,93,93,124,62,
%T A255683 93,93,124,93,124,124,155,63,126,126,189,126,189,189,252,126,189,189,
%U A255683 252,189,252,252,315,126,189,189,252,189,252,252,315,189,252,252,315
%N A255683 Sum of the binary numbers whose digits are cyclic permutations of the binary expansion of n.
%C A255683 All the primes in the sequence are Mersenne primes (A000668).
%H A255683 Paolo P. Lava, <a href="/A255683/b255683.txt">Table of n, a(n) for n = 1..1000</a>
%F A255683 a(2^n) = Sum_{k=1..n} 2^k = 2^(n+1)-1.
%F A255683 a(5+4*k) = a(6+4*k), for k >= 0.
%F A255683 For n >= 0 and 0 <= i <= 2^n - 1 we conjecture a(2^n + i) = (2^(n+1) - 1)*A063787(i+1). An example is given below. - _Peter Bala_, Mar 02 2015
%F A255683 a(n) = A000120(n)*(A000918(A000523(n) + 1) + 1). - _Alan Michael Gómez Calderón_, Jul 07 2025
%e A255683 6 in base 2 is 110 and all the cyclic permutations of its digits are: 110, 101, 011. In base 10 they are 6, 5, 3 and their sum is 6 + 5 + 3 = 14.
%e A255683 From _Peter Bala_, Mar 02 2015: (Start)
%e A255683 Let b(n) = A063787(n), beginning [1, 2, 2, 3, 2, 3, 3, 4, ...]. Then
%e A255683 [a(1)] = 1*[b(1)]; [a(2), a(3)] = 3*[b(1), b(2)];
%e A255683 [a(4), a(5), a(6), a(7)] = 7*[b(1), b(2), b(3), b(4)];
%e A255683 [a(8), a(9), a(10), a(11), a(12), a(13), a(14), a(15)] = 15*[b(1), b(2), b(3), b(4), b(5), b(6), b(7), b(8)].
%e A255683 It is conjectured that this relationship continues. (End)
%p A255683 with(numtheory): P:=proc(q) local a,b,c,k,n;
%p A255683 for n from 1 to q do a:=convert(n,binary,decimal); b:=n; c:=ilog10(a);
%p A255683 for k from 1 to c do a:=(a mod 10)*10^c+trunc(a/10); b:=b+convert(a,decimal,binary); od;
%p A255683 print(b); od; end: P(1000);
%t A255683 f[n_] := Block[{b = 2, w = IntegerDigits[n, b]}, Apply[Plus, FromDigits[#, b] & /@ (RotateRight[w, #] & /@ Range[Length@ w])]]; Array[f, 60] (* _Michael De Vlieger_, Mar 04 2015 *)
%t A255683 Table[Total[FromDigits[#,2]&/@Table[RotateRight[IntegerDigits[k,2],n],{n,IntegerLength[k,2]}]],{k,60}] (* _Harvey P. Dale_, Jan 03 2018 *)
%Y A255683 Cf. A000120, A000225, A000523, A000668, A000918, A063787.
%K A255683 nonn,base,easy
%O A255683 1,2
%A A255683 _Paolo P. Lava_, Mar 02 2015