A 630. Ensembles additivement séparés.
Problème proposé par Michel Lafond
Un ensemble E d’entiers naturels de cardinal N est dit additivement séparé si les 2^N parties de E ont des sommes distinctes deux à deux.
Ainsi E = {1, 2, 4, 8} est additivement séparé car ses 16 parties ont pour sommes les entiers de 0 à 15.
Mais E’ = {3, 5, 6, 7} est aussi additivement séparé car ses 16 parties ont pour sommes distinctes :
0, 3, 5, 6, 7, 3 + 5 = 8, 3 + 6 = 9, 3 + 7 = 10, 5 + 6 = 11, 5 + 7 = 12, 6 + 7 = 13, 3 + 5 + 6 = 14,c3 + 5 + 7
= 15, 3 + 6 + 7 = 16, 5 + 6 + 7 = 18 et 3 + 5 + 6 + 7 = 21, De plus MAX (E’) = 7 est plus petit que MAX (E) = 8.
Trouver pour chaque k∈[2,3,4,5,6,7,8,9,10} un ensemble E_k additivement séparé, ayant k éléments et avec MAX (E_k) minimal.
Solution proposée par Daniel Collignon
Le tableau ci-après décrit les ensembles E
kpour k variant de 2 à 10 E₂ {1,2}
E₃ {2,3,4}
E₄ {3,5,6,7}
E₅ {6,9,11,12,13}
E₆ {11,17,20,22,23,24}
E₇ {20,31,37,40,42,43,44}
E₈ {40,60,71,77,80,82,83,84}
E₉ {77,117,137,148,154,157,159,160,161}
E₁₀ {148,225,265,285,296,302,305,307,308,309}
Ces suites sont mentionnées dans les rubriques de l'OEIS http://oeis.org/A096858 et http://oeis.org/A005318
Par ailleurs, on peut accéder aux articles suivants:
- Tom Bohman A sum packing problem of Erdős and the Conway-Guy sequence à l'adresse:
http://www.ams.org/journals/proc/1996-124-12/S0002-9939-96-03653-2/S0002-9939-96-03653-2.pdf -W.F. Lunnon : Integer sets with distinct subset-sums à l'adresse:
http://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917837-5/S0025-5718-1988-0917837- 5.pdf
-Peter Borwein et Michael J.Mossinghoff : Newman polynomials with prescribed vanisihing and integer sets with distinct subset sums à l'adresse:
http://www.ams.org/journals/mcom/2003-72-242/S0025-5718-02-01460-6/S0025-5718-02-01460-6.pdf
A096858 Triangle read by rows in which row n gives the n-set obtained as the differences {b(n)-b(n-i), 0
<= i <= n-1}, where b() = A005318().
+40 7
1, 1, 2, 2, 3, 4, 3, 5, 6, 7, 6, 9, 11, 12, 13, 11, 17, 20, 22, 23, 24, 20, 31, 37, 40, 42, 43, 44, 40, 60, 71, 77, 80, 82, 83, 84, 77, 117, 137, 148, 154, 157, 159, 160, 161, 148, 225, 265, 285, 296, 302, 305, 307, 308, 309, 285, 433, 510, 550, 570, 581, 587, 590, 592, 593, 594(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
It is conjectured that the triangle has the property that all 2^n subsets of row n have distinct sums. This conjecture was proved by T. Bohman in 1996 - N. J. A. Sloane, Feb 09 2012
It is also conjectured that in some sense this triangle is optimal. See A005318 for further information and additional references.
REFERENCES
J. H. Conway and R. K. Guy, Solution of a problem of Erdős, Colloq. Math. 20 (1969), p. 307.
R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J.
Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982.
R. K. Guy, Unsolved Problems in Number Theory, C8.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
Tom Bohman, A sum packing problem of Erdős and the Conway-Guy sequence, Proc. AMS 124, (No. 12, 1996), pp. 3627-3636.
EXAMPLE
The triangle begins:
{1}
{1,2}
{2,3,4}
{3,5,6,7}
{6,9,11,12,13}
{11,17,20,22,23,24}
{20,31,37,40,42,43,44}
{40,60,71,77,80,82,83,84}
{77,117,137,148,154,157,159,160,161}
{148,225,265,285,296,302,305,307,308,309}
{285,433,510,550,570,581,587,590,592,593,594}
{570,855,1003,1080,1120,1140,1151,1157,1160,1162,1163,1164}
{1120,1690,1975,2123,2200,2240,2260,2271,2277,2280,2282,2283,2284}
{2200,3320,3890,4175,4323,4400,4440,4460,4471,4477,4480,4482,4483,4484}
{4323,6523,7643,8213,8498,8646,8723,8763,8783,8794,8800,8803,8805,8806,8807}
A005318 Conway-Guy sequence: a(n + 1) = 2a(n) - a(n - floor( 1/2 + sqrt(2n) )).
(Formerly M1075)
19
0, 1, 2, 4, 7, 13, 24, 44, 84, 161, 309, 594, 1164, 2284, 4484, 8807, 17305, 34301, 68008, 134852, 267420, 530356, 1051905, 2095003, 4172701, 8311101, 16554194, 32973536, 65679652, 130828948, 261127540, 521203175, 1040311347, 2076449993(list; graph; refs; listen; history;
text; internal format)
OFFSET
0,3
COMMENTS
Conway and Guy conjecture that the set of k numbers {s_i = a(k) - a(k-i) : 1 <= i <= k} has the property that all its subsets have distinct sums - see Guy's book. These k-sets are the rows of A096858. [This conjecture has apparently now been proved by Bohman. - I. Halupczok
(integerSequences(AT)karimmi.de), Feb 20 2006]
REFERENCES
J. H. Conway and R. K. Guy, Solution of a problem of Erdos, Colloq. Math.
20 (1969), p. 307.
R. K. Guy, Unsolved Problems in Number Theory, C8.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Wald, Problem 1192, Unequal sums, J. Rec. Math., 15 (No. 2, 1983-1984), pp. 148-149.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..300
Tom Bohman, A sum packing problem of Erdos and the Conway-Guy sequence, Proc. AMS 124, (No. 12, 1996), pp. 3627-3636.
P. Borwein and M. J. Mossinghoff, Newman Polynomials with Prescribed Vanishing and Integer Sets with Distinct Subset Sums, Math. Comp., 72 (2003), 787-800.
J. H. Conway & R. K. Guy, Sets of natural numbers with distinct sums, Manuscript.
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982.
R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982.
(Annotated scanned copy)
G. Kreweras, Sur quelques problèmes relatifs au vote pondéré [Some
problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.
G. Kreweras, Alvarez Rodriguez, Miguel-Angel, Pondération entière minimale de N telle que pour tout k toutes les k-parties de N aient des poids distincts, [Minimal integer weighting of N such that for any k all the k-subsets of N have unequal weights] C. R. Acad. Sci. Paris Ser. I Math.
296 (1983), no. 8, 345-347.
G. Kreweras, Alvarez Rodriguez, Miguel-Angel, Pondération entière minimale de N telle que pour tout k toutes les k-parties de N aient des poids distincts, [Minimal integer weighting of N such that for any k all the k-subsets of N have unequal weights], C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 8, 345-347. (Annotated scanned copy)
W. F. Lunnon, Integer sets with distinct subset-sums, Math. Comp. 50 (1988), 297-320.
M. Wald & N. J. A. Sloane, Correspondence and Attachment, 1987
