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A145818
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Odd positive integers a(n) such that for every integer m==3(mod 4) there exists a unique representation of the form m=a(l)+2a(s), but there are no such representations for m==1(mod 4)
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11
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1, 5, 17, 21, 65, 69, 81, 85, 257, 261, 273, 277, 321, 325, 337, 341, 1025, 1029, 1041, 1045, 1089, 1093, 1105, 1109, 1281, 1285, 1297, 1301, 1345, 1349, 1361, 1365, 4097, 4101, 4113, 4117, 4161, 4165, 4177, 4181, 4353, 4357, 4369, 4373, 4417, 4421, 4433
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OFFSET
| 1,2
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COMMENTS
| Theorem. A positive odd number is in the sequence iff in its binary expansion all bits in the k-th position from the end, for k=2, 4, 6, ..., are zeros. For example, 337, 341 have binary expansions 101010001, 101010101. Thus both of them are in the sequence. If A(x) is the counting function of a(n)<=x, then A(x)=O(sqrt(x))and Omega(sqrt(x)). If f(x)=sum_{n>=1}x^a(n), abs(x)<1, then f(x)*f(x^2)=x^3/(1-x^4); a(n)=2A145812(n)-1.
Every positive odd integer m==3 (mod 2^(2r)) is a unique sum of the form a(2^(r-1)*(s-1)+1)+a(2^(r-1)*(t-1)+1),r=1,2,..., while other odd integers are not expressible in such form (see also comment to A145812). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 21 2008]
To get the decomposition of m=4k+3 as sum a(l)+2a(s), write m-2 as Sum b_j 2^j, then a(s) = 1 + Sum_{j odd} b_j 2^(j-1). For example, if m=55, then we have: 53=2^0+2^2+2^4+2^5. Thus a(l)=1+2^4 =17 and the required decomposition is: 55=a(l)+2*17,such that a(l)=21. We see that l=4,s=3, i.e. "index coordinates" of 55 are (4,3). Thus we have a one-to-one map of positive integers of the form 4k+3 to the positive lattice points on the plane. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 26 2008]
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LINKS
| Klaus Brockhaus, Table of n, a(n) for n=1..8192 [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 01 2008]
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CROSSREFS
| A145812
Sequence in context: A191210 A191143 A032376 * A029986 A076275 A031270
Adjacent sequences: A145815 A145816 A145817 * A145819 A145820 A145821
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KEYWORD
| nonn
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AUTHOR
| Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 20 2008, Oct 21 2008
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EXTENSIONS
| Extended beyond a(16). Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 22 2008
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