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

Author

Madjid Mirzavaziri, Mar 31 2011

Keywords

• nonn
• nice

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

Examples

			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)
		

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.

Crossrefs

Cf. A188430, A188431.

Cf. A010054, A127648, A002262.

Programs

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

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 *)

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.