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A171886
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Numbers n such that A008949(n) is a power of 2.
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2
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0, 1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 17, 20, 21, 27, 28, 29, 31, 35, 36, 44, 45, 49, 54, 55, 65, 66, 71, 77, 78, 90, 91, 97, 104, 105, 119, 120, 121, 127, 135, 136, 152, 153, 161, 170, 171, 189, 190, 199, 209, 210, 230, 231, 241, 252, 253, 275, 276, 279, 287, 299
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OFFSET
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1,3
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COMMENTS
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Partial sums of binomial coefficients were considered in section 2.2 of the 1964 paper by Leech. The presence of the number 279 corresponds to the existence of the Golay perfect code of length 23.
In general, A000217(n+1)+i-1 is in this sequence IFF the first i items in row n of Pascal's triangle add up to a power of 2.
Almost all members of this sequence are "trivial" terms of four types: A000217(i); A000217(i)+1, A000217(i)+i, and A000217(2i+1)+i for all integers i. 279 is the sole nontrivial term.
The existence of members of this sequence is of course crucial in the study of the existence of perfect binary codes - see the references. - N. J. A. Sloane, Nov 24 2010
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REFERENCES
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M. R. Best, Perfect codes hardly exist. IEEE Trans. Inform. Theory 29 (1983), no. 3, 349-351.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
John Leech, ``Some Sphere Packings in Higher Space'', Can. J. Math., 16 (1964), page 669.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.
A. Tietavainen, On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24 (1973), 88-96.
J. H. van Lint, A survey of perfect codes. Rocky Mountain J. Math. 5 (1975), 199-224.
J. H. van Lint, Recent results on perfect codes and related topics, in Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974), pp. 158-178. Math. Centre Tracts, No. 55, Math. Centrum, Amsterdam, 1974.
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LINKS
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EXAMPLE
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17 is in the sequence because A008949(17)=16, which in turn is because the first 3 elements of row 5 of Pascal's triangle, 1+5+10, add up to 16.
279 is in the sequence because the first 4 elements of row 24 of Pascal's triangle add up to 2^11: 1+23+253+1771=2048.
4097 is in the sequence because the first 3 elements of row 91 of Pascal's triangle add up to 2^12: 1 + 90 + 4005 = 4096. - Reinhard Zumkeller, Aug 08 2013
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PROG
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(Haskell)
import Data.List (elemIndices)
a171886 n = a171886_list !! (n-1)
a171886_list = elemIndices 1 $ map a209229 $ concat a008949_tabl
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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