A004210 "Magic" integers: a(n+1) is the smallest integer m such that there is no overlap between the sets {m, m-a(i), m+a(i): 1 <= i <= n} and {a(i), a(i)-a(j), a(i)+a(j): 1 <= j < i <= n}. (Formerly M2728)

1, 3, 8, 18, 30, 43, 67, 90, 122, 161, 202, 260, 305, 388, 416, 450, 555, 624, 730, 750, 983, 1059, 1159, 1330, 1528, 1645, 1774, 1921, 2140, 2289, 2580, 2632, 2881, 3158, 3304, 3510, 3745, 4086, 4563, 4741, 4928, 5052, 5407, 5864, 6242, 6528, 6739, 7253

Offset

1,2

Comments

The definition implies that the sets {a(i)} (A004210), {a(i)-a(j), j < i} (A206522) and {a(i)+a(j), j < i} (A206523) are disjoint. A206524 gives the complement of their union.

References

R. A. Bates, E. Riccomagno, R. Schwabe, H. P. Wynn, Lattices and dual lattices in optimal experimental design for Fourier models, Computational Statistics & Data Analysis Volume 28, Issue 3, 4 September 1998, Pages 283-296. See page 293.

D. R. Hofstadter, "Goedel, Escher, Bach: An Eternal Golden Braid", Basic Books Incorporated, p. 73

P. Mark Kayll, Well-spread sequences and edge-labelings with constant Hamiltonian weight, Disc. Math. & Theor. Comp. Sci 6 2 (2004) 401-408

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..150

B. G. DeBoer, Letter to N. J. A. Sloane, Dec 15 1978, with enclosure of Silvertom article.

J. V. Silverton, On the generation of 'magic integrals', Acta Cryst. A34 (1978) p. 634.

Eric Weisstein's World of Mathematics, Magic Integer.

Formula

a(n+1) = min{ k | k and k +- a(i) are not equal to a(i) or a(i)-a(j) or a(i)+a(j) for any n+1 > i > j > 0}. [Corrected by T. D. Noe, Sep 08 2008]

Mathematica

a[1] = 1; a[n_] := a[n] = Module[{pairs = Flatten[ Table[{a[j] + a[k], a[k] - a[j]}, {j, 1, n-1}, {k, j+1, n-1}]], an = a[n-1] + 1}, While[ True, If[ Intersection[ Join[ Array[a, n-1], pairs], Prepend[ Flatten[ Table[{a[j] + an, an - a[j]}, {j, 1, n-1}]], an]] == {}, Break[], an++]]; an]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Nov 10 2011 *)

Prog

(Haskell)
import Data.List (intersect)
a004210 n = a004210_list !! (n-1)
a004210_list = magics 1 [0] [0] where
   magics :: Integer -> [Integer] -> [Integer] -> [Integer]
   magics n ms tests
      | tests `intersect` nMinus == [] && tests `intersect` nPlus == []
      = n : magics (n+1) (n:ms) (nMinus ++ nPlus ++ tests)
      | otherwise
      = magics (n+1) ms tests
      where nMinus = map (n -) ms
            nPlus  = map (n +) ms
-- magics works also as generator for a126428_list, cf. A126428.
-- Reinhard Zumkeller, Mar 03 2011

Crossrefs

Cf. A000969, A005228, A206522, A206523, A206524.

Sequence in context: A171701 A288249 A360139 * A247022 A119881 A184636

Adjacent sequences: A004207 A004208 A004209 * A004211 A004212 A004213

Keyword

easy,nonn,nice

Author

N. J. A. Sloane, following a suggestion from B. G. DeBoer, Dec 15 1978

Extensions

Additional comments from Robert M. Burton, Jr. (bob(AT)oregonstate.edu), Feb 20 2005

More terms from Joshua Zucker, May 04 2006

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

Edited by N. J. A. Sloane, Feb 08 2012

Status

approved