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A178207
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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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1
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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MATHEMATICA
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Table[4 n - 2 Total@ IntegerDigits[n, 2] + 1, {n, 0, 59}] (* Michael De Vlieger, Jan 02 2016 *)
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PROG
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(PARI) a(n) = 4*n - 2*hammingweight(n) + 1; \\ Michel Marcus, Jan 02 2016
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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17 inserted and sequence extended by R. J. Mathar, May 24 2010
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STATUS
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approved
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