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A304935 a(n) is the largest possible integer value for sqrt(0 1 2 ... n), where one is allowed to place any mixture of +'s and *'s in the n blank spaces.

%I A304935 #19 Jun 25 2018 02:57:35
%S A304935 1,0,0,5,11,6,71,19,123,33,174,426,174,233,625,816,5695,3656,15936,
%T A304935 246960,24234,24234,35151,140604,177399,250982,1304130,1304130,
%U A304935 1304130,1304130,5532955,5532955,58136459,8525544,8525544,58136459,941988492,58136459,941988492
%N A304935 a(n) is the largest possible integer value for sqrt(0 _ 1 _ 2 _ ... _ n), where one is allowed to place any mixture of +'s and *'s in the n blank spaces.
%C A304935 Inspired by a test ARML problem from 2018, which asked students to determine a(8).
%H A304935 Rémy Sigrist, <a href="/A304935/a304935.gp.txt">PARI program for A304935</a>
%e A304935 a(2) = a(3) = 0, since no positive squares are achievable.
%e A304935 Some examples:
%e A304935 a(7) = 71 = sqrt(0+1+2*3*4*5*6*7).
%e A304935 a(8) = 19 = sqrt(0+1*2+3+4*5+6*7*8).
%e A304935 a(20) = 246960 = sqrt(0+1*2*3*4*5*6*7*8*9*10*11+12*13*14*15*16*17*18*19*20)
%t A304935 sqStrTest[n_] := Module[{bVal, bStr, i, j, iB, mVal, mStr},
%t A304935   bVal = -1;
%t A304935   For[i = 0, i < 2^n, i++,
%t A304935    iB = IntegerDigits[i, 2];
%t A304935    While[Length[iB] < n, PrependTo[iB, 0]];
%t A304935    mStr = "0";
%t A304935    For[j = 1, j <= n, j++,
%t A304935     mStr = StringJoin[mStr, If[iB[[j]] == 0, "+", "*"], ToString[j]]];
%t A304935    mVal = ToExpression[mStr];
%t A304935    If[Sqrt[mVal] == Floor[Sqrt[mVal]],
%t A304935     If[mVal > bVal, {bVal, bStr} = {mVal, mStr}]
%t A304935     ]
%t A304935    ];
%t A304935   Print[{Sqrt[bVal], bVal, bStr}]]
%o A304935 (PARI) See Links section.
%Y A304935 Upper-bounded by sqrt(A038507).
%K A304935 nonn
%O A304935 1,4
%A A304935 _Andy Niedermaier_, May 21 2018
%E A304935 More terms from _Rémy Sigrist_, May 22 2018