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a(n) = 4*n - 2*A000120(n) + 1 where A000120(n) = number of nonzero digits in the binary representation of n.
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%I #25 Feb 12 2016 17:21:12

%S 1,3,7,9,15,17,21,23,31,33,37,39,45,47,51,53,63,65,69,71,77,79,83,85,

%T 93,95,99,101,107,109,113,115,127,129,133,135,141,143,147,149,157,159,

%U 163,165,171,173,177,179,189,191,195,197,203,205,209,211,219,221,225,227

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

%C 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.

%H Robert Israel, <a href="/A178207/b178207.txt">Table of n, a(n) for n = 0..10000</a>

%H Djordje Baralic, Branislav Prvulovic, Gordana Stojanovic, Sinisa Vrecica, Rade Zivaljevic, <a href="http://arxiv.org/abs/1005.3709">Topological obstructions to totally skew embeddings</a>, arXiv:1005.3709 [math.AT], 2010.

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

%F a(0) = 1; a(n) = 2n + a(floor(n/2)). - _Daniel Suteu_, Jan 01 2016

%F G.f. A(x) satisfies A(x) = 2*x/(1-x)^2 + (1+x)*A(x^2). - _Robert Israel_, Jan 01 2016

%e a(0) = 4*1 - 2*A000120(0) + 1 = (4*0)-(2*0)+1 = 1.

%e a(1) = 4*1 - 2*A000120(1) + 1 = (4*1)-(2*1)+1 = 3 is prime.

%e a(2) = 4*2 - 2*A000120(2) + 1 = (4*2)-(2*1)+1 = 7 is prime.

%e a(3) = 4*3 - 2*A000120(3) + 1 = (4*3)-(2*2)+1 = 9.

%e a(4) = 4*4 - 2*A000120(4) + 1 = (4*4)-(2*1)+1 = 15.

%e a(5) = 4*5 - 2*A000120(5) + 1 = (4*5)-(2*2)+1 = 17 is prime.

%p 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

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

%o (PARI) a(n) = 4*n - 2*hammingweight(n) + 1; \\ _Michel Marcus_, Jan 02 2016

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

%K easy,nonn

%O 0,2

%A _Jonathan Vos Post_, May 22 2010

%E 17 inserted and sequence extended by _R. J. Mathar_, May 24 2010