A188429 L(n) is the minimum of the largest elements of all n-full sets, or 0 if no such set exists.

1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 9, 9, 9, 10, 10, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 12, 13, 13

Offset

1,3

Comments

Let A be a set of positive integers. We say that A is n-full if (sum A)=[n] for a positive integer n, where (sum A) is the set of all positive integers which are a sum of distinct elements of A and [n]={1,2,...,n}. The number L(n) denotes the minimum of the set {max A: (sum A)=[n] }.

Terms m > 7 occur exactly m times. - Reinhard Zumkeller, Aug 06 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Mohammad Saleh Dinparvar, Python program

L. Naranjani and M. Mirzavaziri, Full Subsets of N, Journal of Integer Sequences, 14 (2011), Article 11.5.3.

Formula

for n>= 15. Let n=k(k+1)/2+r, where r=0,1,..., k then

|k, if r=0

L(n) = |k+1, if 1 <= r <= k-2

|k+2, if k-1 <= r <= k.

Example

From Reinhard Zumkeller, Aug 06 2015: (Start)
Compressed table: no commas and for a and k: 10 replaced by A, 11 by B.
. -----------------------------------------------------------------------------
.   n   1   5   10   15   20   25   30   35   40   45   50   55   60   65   70
. ----  .---.----.----.----.----.----.----.----.----.----.----.----.----.----.-
. t(n)  10100100010000100000100000010000000100000000100000000010000000000100000
. k(n)  1 2  3   4    5     6      7       8        9         A          B
. r(n)  0101201230123401234501234560123456701234567801234567890123456789A012345
. ----  -----------------------------------------------------------------------
. a(n)  102003400455675666776777788788888998999999AA9AAAAAAABBABBBBBBBBCCBCCCCC
. -----------------------------------------------------------------------------
where t(n)=A010054(n), k(n)=A127648(n) zeros blanked, and r(n)=A002262(n). (End)

Mathematica

kr[n_] := {k, r} /. ToRules[Reduce[0 <= r <= k && n == k*((k+1)/2)+r, {k, r}, Integers]]; L[n_] := Which[{k0, r0} = kr[n]; r0 == 0, k0, 1 <= r0 <= k0-2, k0+1, k0-1 <= r0 <= k0, k0+2]; Join[{1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7}, Table[L[n], {n, 15, 80}]] (* Jean-François Alcover, Oct 10 2015 *)

Prog

(Haskell)
a188429 n = a188429_list !! (n-1)
a188429_list = [1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7] ++
               f [15 ..] (drop 15 a010054_list) 0 4
   where f (x:xs) (t:ts) r k | t == 1    = (k + 1) : f xs ts 1 (k + 1)
                             | r < k - 1 = (k + 1) : f xs ts (r + 1) k
                             | otherwise = (k + 2) : f xs ts (r + 1) k
-- Reinhard Zumkeller, Aug 06 2015

Crossrefs

Cf. A188430, A188431.

Cf. A010054, A127648, A002262.

Sequence in context: A074734 A174956 A124182 * A188430 A013585 A261319

Adjacent sequences: A188426 A188427 A188428 * A188430 A188431 A188432

Keyword

nonn,nice

Author

Madjid Mirzavaziri, Mar 31 2011

Status

approved