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(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
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