A188429 - OEIS

%I #37 May 11 2020 02:38:51

%S 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,

%T 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,

%U 11,11,11,12,12,11,12,12,12,12,12,12,12,12,12,13,13,12,13,13

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

%C 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] }.

%C Terms m > 7 occur exactly m times. - _Reinhard Zumkeller_, Aug 06 2015

%H Reinhard Zumkeller, <a href="/A188429/b188429.txt">Table of n, a(n) for n = 1..10000</a>

%H Mohammad Saleh Dinparvar, <a href="http://github.com/SalehDinparvar/sequence_computer/blob/master/A188429.py">Python program</a>

%H L. Naranjani and M. Mirzavaziri, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Mirzavaziri/mirza4.html">Full Subsets of N</a>, Journal of Integer Sequences, 14 (2011), Article 11.5.3.

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

%F        |k, if r=0

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

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

%e From _Reinhard Zumkeller_, Aug 06 2015: (Start)

%e Compressed table: no commas and for a and k: 10 replaced by A, 11 by B.

%e . -----------------------------------------------------------------------------

%e .   n   1   5   10   15   20   25   30   35   40   45   50   55   60   65   70

%e . ----  .---.----.----.----.----.----.----.----.----.----.----.----.----.----.-

%e . t(n)  10100100010000100000100000010000000100000000100000000010000000000100000

%e . k(n)  1 2  3   4    5     6      7       8        9         A          B

%e . r(n)  0101201230123401234501234560123456701234567801234567890123456789A012345

%e . ----  -----------------------------------------------------------------------

%e . a(n)  102003400455675666776777788788888998999999AA9AAAAAAABBABBBBBBBBCCBCCCCC

%e . -----------------------------------------------------------------------------

%e where t(n)=A010054(n), k(n)=A127648(n) zeros blanked, and r(n)=A002262(n). (End)

%t 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 *)

%o (Haskell)

%o a188429 n = a188429_list !! (n-1)

%o a188429_list = [1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7] ++

%o                f [15 ..] (drop 15 a010054_list) 0 4

%o    where f (x:xs) (t:ts) r k | t == 1    = (k + 1) : f xs ts 1 (k + 1)

%o                              | r < k - 1 = (k + 1) : f xs ts (r + 1) k

%o                              | otherwise = (k + 2) : f xs ts (r + 1) k

%o -- _Reinhard Zumkeller_, Aug 06 2015

%Y Cf. A188430, A188431.

%Y Cf. A010054, A127648, A002262.

%K nonn,nice

%O 1,3

%A _Madjid Mirzavaziri_, Mar 31 2011