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A255683
Sum of the binary numbers whose digits are cyclic permutations of the binary expansion of n
1
1, 3, 6, 7, 14, 14, 21, 15, 30, 30, 45, 30, 45, 45, 60, 31, 62, 62, 93, 62, 93, 93, 124, 62, 93, 93, 124, 93, 124, 124, 155, 63, 126, 126, 189, 126, 189, 189, 252, 126, 189, 189, 252, 189, 252, 252, 315, 126, 189, 189, 252, 189, 252, 252, 315, 189, 252, 252, 315
OFFSET
1,2
COMMENTS
a(2^n) = Sum_{k=1..n} 2^k = 2^(n+1)-1.
a(5+4*k) = a(6+4*k), for k >= 0.
All the primes in the sequence are Mersenne primes (A000668).
LINKS
FORMULA
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
EXAMPLE
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.
From Peter Bala, Mar 02 2015: (Start)
Let b(n) = A063787(n), beginning [1, 2, 2, 3, 2, 3, 3, 4, ...]. Then
[a(1)] = 1*[b(1)]; [a(2), a(3)] = 3*[b(1), b(2)];
[a(4), a(5), a(6), a(7)] = 7*[b(1), b(2), b(3), b(4)];
[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)].
It is conjectured that this relationship continues. (End)
MAPLE
with(numtheory): P:=proc(q) local a, b, c, k, n;
for n from 1 to q do a:=convert(n, binary, decimal); b:=n; c:=ilog10(a);
for k from 1 to c do a:=(a mod 10)*10^c+trunc(a/10); b:=b+convert(a, decimal, binary); od;
print(b); od; end: P(1000);
MATHEMATICA
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 *)
Table[Total[FromDigits[#, 2]&/@Table[RotateRight[IntegerDigits[k, 2], n], {n, IntegerLength[k, 2]}]], {k, 60}] (* Harvey P. Dale, Jan 03 2018 *)
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Paolo P. Lava, Mar 02 2015
STATUS
approved