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

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

Offset: 1

Author

Robert Munafo, Oct 16 2010

Keywords

• nonn

Comments

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

a(230) = 4097 is another nontrivial term, see example. - Reinhard Zumkeller, Aug 08 2013

Examples

			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
		

References

• 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.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• John Leech, Some Sphere Packings in Higher Space (PDF available from the publisher).

Crossrefs

Cf. A008949.

Cf. A209229.

Programs

Haskell

import Data.List (elemIndices)
a171886 n = a171886_list !! (n-1)
a171886_list = elemIndices 1 $ map a209229 $ concat a008949_tabl
-- Reinhard Zumkeller, Aug 08 2013

Extensions

Edited by N. J. A. Sloane, Oct 18 2010

Offset changed by Reinhard Zumkeller, Aug 08 2013