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

1, 4, 9, 16, 36, 81, 144, 324, 625, 1156, 2401, 4900, 9801, 19600, 39204, 78400, 156816, 313600, 627264, 1254400, 2509056, 5022081, 10042561, 20088324, 40182921, 80371225, 160731684, 321484900, 642977449, 1285939600

Offset

1,2

Comments

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.

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.

Links

Table of n, a(n) for n=1..30.

Example

Up to a(4)=16, we have a(n)=n^2.
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.
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.
But in contrast to A064934, such a simple formula (with equality) cannot be used here:
a(7)=12^2=144 < 147=sum(a(k),k<7) and also a(10)=sum(a(k),k<10)-84.

Prog

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

Crossrefs

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

Sequence in context: A356880 A018228 A204503 * A076967 A233247 A363657

Adjacent sequences: A138855 A138856 A138857 * A138859 A138860 A138861

Keyword

nonn

Author

M. F. Hasler, Apr 09 2008

Extensions

a(24)-a(30) from Donovan Johnson, Oct 03 2009

Status

approved