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A188429 L(n) is the minimum of the largest elements of all n-full sets, or 0 if no such set exists. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois 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

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Last modified December 7 03:00 EST 2019. Contains 329836 sequences. (Running on oeis4.)