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A171886 Numbers n such that A008949(n) is a power of 2. 2


%S 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,

%T 65,66,71,77,78,90,91,97,104,105,119,120,121,127,135,136,152,153,161,

%U 170,171,189,190,199,209,210,230,231,241,252,253,275,276,279,287,299

%N Numbers n such that A008949(n) is a power of 2.

%C 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.

%C 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.

%C 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.

%C 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

%C a(230) = 4097 is another nontrivial term, see example. - _Reinhard Zumkeller_, Aug 08 2013

%D M. R. Best, Perfect codes hardly exist. IEEE Trans. Inform. Theory 29 (1983), no. 3, 349-351.

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.

%D John Leech, ``Some Sphere Packings in Higher Space'', Can. J. Math., 16 (1964), page 669.

%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.

%D A. Tietavainen, On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24 (1973), 88-96.

%D J. H. van Lint, A survey of perfect codes. Rocky Mountain J. Math. 5 (1975), 199-224.

%D 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.

%H Reinhard Zumkeller, <a href="/A171886/b171886.txt">Table of n, a(n) for n = 1..10000</a>

%H John Leech, <a href="http://www.cms.math.ca/cjm/v16/cjm1964v16.0657-0682.pdf">Some Sphere Packings in Higher Space</a> (PDF available from the publisher).

%e 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.

%e 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.

%e 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

%o (Haskell)

%o import Data.List (elemIndices)

%o a171886 n = a171886_list !! (n-1)

%o a171886_list = elemIndices 1 $ map a209229 $ concat a008949_tabl

%o -- _Reinhard Zumkeller_, Aug 08 2013

%Y Cf. A008949.

%Y Cf. A209229.

%K nonn

%O 1,3

%A _Robert Munafo_, Oct 16 2010

%E Edited by _N. J. A. Sloane_, Oct 18 2010

%E Offset changed by _Reinhard Zumkeller_, Aug 08 2013

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Last modified April 12 11:49 EDT 2021. Contains 342920 sequences. (Running on oeis4.)