login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A210380 Consider all n-tuples of distinct positive integers for which no two different elements add up to a square. This sequence gives the smallest maximal integer in such tuples. 4
1, 2, 4, 6, 9, 10, 11, 15, 18, 20, 21, 24, 26, 28, 32, 34, 36, 38, 40, 42, 50, 52, 54, 56, 58, 60, 62, 64, 72, 74, 76, 78, 80, 82, 84, 86, 88, 99, 101, 103, 105, 107, 109, 111, 114, 116, 118, 129, 130, 133, 135, 137, 139, 141, 143, 145, 152, 159, 160, 163, 167 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

J. P. Massias, Sur les suites dont les sommes des termes 2 à 2 ne sont pas des carrés, Publications du département de mathématiques de Limoges, 1982.

LINKS

Jon E. Schoenfield and Giovanni Resta, Table of n, a(n) for n = 1..175 (first 100 terms from Jon E. Schoenfield)

Ayman Khalfalah, Sachin Lodha, and Endre Szemerédi, Tight bound for the density of sequence of integers the sum of no two of which is a perfect square, Discr. Math. 256 (2002) 243 [DOI]

J. C. Lagarias, A. M. Odlyzko, J. B. Shearer, On the density of sequences of integers the sum of no two of which is a square. I. Arithmetic progressions, Journal of Combinatorial Theory. Series A, 33 (1982), pp. 167-185.

J. C. Lagarias, A. M. Odlyzko, J. B. Shearer, On the density of sequences of integers the sum of no two of which is a square. II. General sequences, Journal of Combinatorial Theory. Series A, 34 (1983), pp. 123-139.

Jon E. Schoenfield, Lexicographically first sequences for n = 1..100

Jon E. Schoenfield, Excel/VBA macro

FORMULA

a(n) ~ (32/11)n.

a(n) <= (32/11)n - 2. Erdos conjectures that a(n) >= (32/11)n - k for some fixed k.

EXAMPLE

For a(29)=72 one sequence is 8, 10, 12, 14, 19, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 61, 63, 65, 72. - Giovanni Resta, Dec 24 2012

The above example sequence is the lexicographically first 29-tuple of distinct positive integers for which no two different elements add up to a square and the maximal integer is a(29). For such sequences for a(1)..a(100), see the "Lexicographically first sequences for n = 1..100" link. - Jon E. Schoenfield, Jan 31 2014

MATHEMATICA

CZ[v_List] :=

   Block[{u = Most[v]}, If[Length[u] > 0 && Last[u] == 0, CZ[u], u]]

ev[v_List] := ev[v] =

   Module[{h = Plus @@ v, u = v}, If[h < 2, h, h = ev[CZ[u]];

    For[k = Floor[Sqrt[Length[u]]] + 1, k < Sqrt[2*Length[u]], k++,

     u[[k^2 - Length[u]]] = 0]; Max[h, 1 + ev[CZ[u]]]]]

a[n_] := Module[{k = n, t}, While[True, t = ev[Table[1, {k}]];

   If[t == n, Return[k], k += n - t]]]

PROG

(PARI) most(v)=my(h=sum(i=1, #v, v[i]), m, u); if(h<2, return(h)); m=#v; while(v[m]==0, m--); u=vector(m-1, i, v[i]); h=most(u); for(k=sqrtint(m)+1, sqrtint(2*m-1), u[k^2-m]=0); max(h, 1+most(u))

a(n)=my(k=n, t); while(1, t=most(vector(k, i, 1)); if(t==n, return(k)); k+=n-t)

CROSSREFS

See A099107 for another version.

Sequence in context: A114526 A178126 A162202 * A228359 A156165 A024968

Adjacent sequences:  A210377 A210378 A210379 * A210381 A210382 A210383

KEYWORD

nonn,nice

AUTHOR

Charles R Greathouse IV, Mar 27 2012

EXTENSIONS

a(25)-a(29) from Giovanni Resta, Dec 24 2012

More terms from Jon E. Schoenfield, Dec 28 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 18 04:44 EST 2019. Contains 329248 sequences. (Running on oeis4.)