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

1,3

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

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).

Example

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

Prog

(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

Crossrefs

Cf. A008949.

Cf. A209229.

Sequence in context: A062470 A179460 A344281 * A343238 A018559 A057196

Adjacent sequences: A171883 A171884 A171885 * A171887 A171888 A171889

Keyword

nonn

Author

Robert Munafo, Oct 16 2010

Extensions

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

Offset changed by Reinhard Zumkeller, Aug 08 2013

Status

approved