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a(n) = minimum (floor(n^3/m) + m) for any integer m >= 1.
2

%I #22 Jun 25 2019 01:07:02

%S 2,5,10,16,22,29,37,45,54,63,72,83,93,104,116,128,140,152,165,178,192,

%T 206,220,235,250,265,280,296,312,328,345,362,379,396,414,432,450,468,

%U 487,505,525,544,563,583,603,623,644,665,686,707,728,749,771,793,815,838,860,883,906,929,952,976,1000

%N a(n) = minimum (floor(n^3/m) + m) for any integer m >= 1.

%H Robert Israel, <a href="/A135061/b135061.txt">Table of n, a(n) for n = 1..10000</a>

%H Robert Israel, <a href="/A135061/a135061.pdf">A135061(n) = floor(2 n^(3/2))</a>

%F a(n) = floor(2*n^(3/2)), with the minimum occurring at m = ceiling(n^(3/2)). See link for proof. - _Robert Israel_, Mar 03 2017

%p f:= proc(n) local m,U,L;

%p U:= floor(2*n^(3/2));

%p m:=floor(n^(3/2));

%p L:= floor(min(n^3/m+m, n^3/(m+1)+m+1));

%p if L <> U then error("Conjecture fails") fi;

%p L

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Mar 01 2017

%o (PARI) a(n)=local( minsum=0, cursum =0, minm=0, lastminsum=0); minsum = n^3 + 1; lastminsum= n^3 + 1; minm =1; for(m=1,n^3, cursum = floor(n^3/m + m); lastminsum = minsum; if(cursum < minsum,minsum = cursum); if(cursum < lastminsum,minm=m); ); print1("minm="minm" "); print1("minsum="minsum" ");

%Y Cf. A135072.

%K nonn

%O 1,1

%A _Alexander R. Povolotsky_, Feb 11 2008, Feb 15 2008

%E a(2) corrected and more terms added by _Robert Israel_, Mar 01 2017