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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 (list; graph; refs; listen; history; text; internal format)
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

Paolo P. Lava, Table of n, a(n) for n = 1..1000

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

Cf. A000225, A000668, A063787.

Sequence in context: A245394 A137473 A303604 * A127307 A099403 A037015

Adjacent sequences:  A255680 A255681 A255682 * A255684 A255685 A255686

KEYWORD

nonn,base,easy

AUTHOR

Paolo P. Lava, Mar 02 2015

STATUS

approved

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Last modified November 21 09:13 EST 2018. Contains 317431 sequences. (Running on oeis4.)