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 A145818 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) + 2*a(s), but there are no such representations for m == 1 (mod 4). 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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). [Vladimir Shevelev, 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. [Vladimir Shevelev, Oct 26 2008] LINKS Klaus Brockhaus, Table of n, a(n) for n=1..8192 [From Klaus Brockhaus, Nov 01 2008] MATHEMATICA aQ[n_] := OddQ[n] && Module[{d = Reverse[IntegerDigits[n, 2]]}, Length[d] < 2 || Max[d[[2;; -1;; 2]]] == 0]; Select[Range[4500], aQ] (* Amiram Eldar, Dec 15 2018 *) PROG (PARI) isok(n) = {if (n % 2, my(rb = Vecrev(binary(n)), brb = vector(#rb\2, k, rb[2*k])); (#brb == 0) || vecmax(brb) == 0, 0); } \\ Michel Marcus, Dec 15 2018 CROSSREFS Cf. A145812. Sequence in context: A191210 A191143 A032376 * A029986 A076275 A316307 Adjacent sequences:  A145815 A145816 A145817 * A145819 A145820 A145821 KEYWORD nonn AUTHOR Vladimir Shevelev, Oct 20 2008, Oct 21 2008 EXTENSIONS Extended beyond a(16) by Klaus Brockhaus, Oct 22 2008 STATUS approved

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Last modified April 20 04:42 EDT 2019. Contains 322294 sequences. (Running on oeis4.)