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 A201052 a(n) is the maximal number c of integers that can be chosen from {1,2,...,n} so that all 2^c subsets have distinct sums. 3
 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In the count 2^c of the cardinality of subsets of a set with cardinality c, the empty set - with sum 0 - is included; 2^c is just the row sum of the c-th row in the Pascal triangle. Conjecture (confirmed through k=7): a(n)=k for all n in the interval A005318(k) <= n < A005318(k+1). - Jon E. Schoenfield, Nov 28 2013 [Note: This conjecture is false; see A276661 for a counterexample (n=34808712605260918463) in which n is in the interval A005318(66) <= n < A005318(67), yet a(n)=67, not 66. - Jon E. Schoenfield, Nov 05 2016] LINKS Jon E. Schoenfield, Table of n, a(n) for n = 1..120 T. Khovanova, The weight puzzle sequence, SeqFan Mailing list Aug 24 2010 T. Khovanova et al., The weights puzzle Jon E. Schoenfield, Excel/VBA macro EXAMPLE Numbers n and an example of a subset of {1..n} exhibiting the maximum cardinality c=a(n): 1, {1} 2, {1, 2} 3, {1, 2} 4, {1, 2, 4} 5, {1, 2, 4} 6, {1, 2, 4} 7, {3, 5, 6, 7} 8, {1, 2, 4, 8} 9, {1, 2, 4, 8} 10, {1, 2, 4, 8} 11, {1, 2, 4, 8} 12, {1, 2, 4, 8} 13, {3, 6, 11, 12, 13} 14, {1, 6, 10, 12, 14} 15, {1, 6, 10, 12, 14} 16, {1, 2, 4, 8, 16} 17, {1, 2, 4, 8, 16} 18, {1, 2, 4, 8, 16} For examples of maximum-cardinality subsets at values of n where a(n) > a(n-1), see A096858. - Jon E. Schoenfield, Nov 28 2013 MAPLE # is any subset of L uniquely determined by its total weight? iswts := proc(L)     local wtset, s, c, subL, thiswt ;     # the weight sums are to be unique, so sufficient to remember the set     wtset := {} ;     # loop over all subsets of weights generated by L     for s from 1 to nops(L) do         c := combinat[choose](L, s) ;         for subL in c do             # compute the weight sum in this subset             thiswt := add(i, i=subL) ;             # if this weight sum already appeared: not a candidate             if thiswt in wtset then                 return false;             else                 wtset := wtset union {thiswt} ;             end if;         end do:     end do:     # All different subset weights were different: success     return true; end proc: # main sequence: given grams 1 to n, determine a subset L # such that each subset of this subset has a different sum. wts := proc(n)     local s, c, L ;     # select sizes from n (largest size first) down to 1,     # so the largest is detected first as required by the puzzle.     for s from n to 1 by -1 do         # all combinations of subsets of s different grams         c := combinat[choose]([seq(i, i=1..n)], s) ;         for L in c do             # check if any of these meets the requir, print if yes             # and return             if iswts(L) then                 print(n, L) ;                 return nops(L) ;             end if;         end do:     end do:     print(n, "-") ; end proc: # loop for weights with maximum n for n from 1 do     wts(n) ; end do: # R. J. Mathar, Aug 24 2010 CROSSREFS Cf. A005318, A096858, A275972, A276661. Sequence in context: A085727 A143442 A137300 * A278044 A255121 A095791 Adjacent sequences:  A201049 A201050 A201051 * A201053 A201054 A201055 KEYWORD nonn,nice AUTHOR N. J. A. Sloane, Nov 26 2011 EXTENSIONS More terms from Alois P. Heinz, Nov 27 2011 More terms from Jon E. Schoenfield, Nov 28 2013 STATUS approved

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Last modified October 17 11:59 EDT 2019. Contains 328110 sequences. (Running on oeis4.)