A178207 a(n) = 4*n - 2*A000120(n) + 1 where A000120(n) = number of nonzero digits in the binary representation of n.

1, 3, 7, 9, 15, 17, 21, 23, 31, 33, 37, 39, 45, 47, 51, 53, 63, 65, 69, 71, 77, 79, 83, 85, 93, 95, 99, 101, 107, 109, 113, 115, 127, 129, 133, 135, 141, 143, 147, 149, 157, 159, 163, 165, 171, 173, 177, 179, 189, 191, 195, 197, 203, 205, 209, 211, 219, 221, 225, 227

Offset

0,2

Comments

Baralic provides some evidence for the conjecture that each n-dimensional, compact smooth manifold M^n (n>1), admits a totally skew embedding in the Euclidean space of dimension N = 4n - 2alpha(n) + 1 where alpha(n) = number of nonzero digits in the binary representation of n. The subsequence of primes begins: 3, 7, 23, 31, 37, 47, 53, 71, 79, 83, 101, 107, 109, 113, 127, 149, 157. See also my note on the Whitney immersion theorem in A005187.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Djordje Baralic, Branislav Prvulovic, Gordana Stojanovic, Sinisa Vrecica, Rade Zivaljevic, Topological obstructions to totally skew embeddings, arXiv:1005.3709 [math.AT], 2010.

Formula

a(n) = A008586(n) - 2*A000120(n) + 1.

a(0) = 1; a(n) = 2n + a(floor(n/2)). - Daniel Suteu, Jan 01 2016

G.f. A(x) satisfies A(x) = 2*x/(1-x)^2 + (1+x)*A(x^2). - Robert Israel, Jan 01 2016

Example

a(0) = 4*1 - 2*A000120(0) + 1 = (4*0)-(2*0)+1 = 1.
a(1) = 4*1 - 2*A000120(1) + 1 = (4*1)-(2*1)+1 = 3 is prime.
a(2) = 4*2 - 2*A000120(2) + 1 = (4*2)-(2*1)+1 = 7 is prime.
a(3) = 4*3 - 2*A000120(3) + 1 = (4*3)-(2*2)+1 = 9.
a(4) = 4*4 - 2*A000120(4) + 1 = (4*4)-(2*1)+1 = 15.
a(5) = 4*5 - 2*A000120(5) + 1 = (4*5)-(2*2)+1 = 17 is prime.

Maple

read("transforms") ; A178207 := proc(n) 4*n-2*wt(n)+1 ; end proc: seq(A178207(n), n=0..120) ; # R. J. Mathar, May 24 2010

Mathematica

Table[4 n - 2 Total@ IntegerDigits[n, 2] + 1, {n, 0, 59}] (* Michael De Vlieger, Jan 02 2016 *)

Prog

(PARI) a(n) = 4*n - 2*hammingweight(n) + 1; \\ Michel Marcus, Jan 02 2016

Crossrefs

Cf. A000120, A005187, A008586, A059997, A124153.

Sequence in context: A279617 A199919 A346303 * A210532 A110872 A032424

Adjacent sequences: A178204 A178205 A178206 * A178208 A178209 A178210

Keyword

easy,nonn

Author

Jonathan Vos Post, May 22 2010

Extensions

17 inserted and sequence extended by R. J. Mathar, May 24 2010

Status

approved