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A138858 a(n)=least square such that the subsets of {a(1),...,a(n)} sum to 2^n different values. 2

%I

%S 1,4,9,16,36,81,144,324,625,1156,2401,4900,9801,19600,39204,78400,

%T 156816,313600,627264,1254400,2509056,5022081,10042561,20088324,

%U 40182921,80371225,160731684,321484900,642977449,1285939600

%N a(n)=least square such that the subsets of {a(1),...,a(n)} sum to 2^n different values.

%C Asking for 2^n different values implies that a(n) differs from all a(k), k<n and in view of the minimality condition, also that a(n) > a(n-1) for n>1.

%C Note that a(n) is close to, but not always larger than sum(a(k),k=1..n-1), as opposed to the case in A064934.

%e Up to a(4)=16, we have a(n)=n^2.

%e But since 5^2=25=9+16 is already represented as sum of earlier terms, this is excluded, while a(5)=6^2=36 has the required property.

%e Obviously, any square larger to the sum of all preceding terms leads to enough new terms, thus a(n) <= floor( sqrt( sum(a(k),k=1..n-1))+1)^2.

%e But in contrast to A064934, such a simple formula (with equality) cannot be used here:

%e a(7)=12^2=144 < 147=sum(a(k),k<7) and also a(10)=sum(a(k),k<10)-84.

%o (PARI) {s=1;p=0; for( n=1,20, until( !bitand( s, s>>(p^2) ), p++); s+=s<<(p^2); print1( p^2,","))}

%Y Cf. A138857 (=sqrt(a(n))), A138000-138001, A064934.

%K nonn

%O 1,2

%A _M. F. Hasler_, Apr 09 2008

%E a(24)-a(30) from _Donovan Johnson_, Oct 03 2009

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)