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.

1, 0, 0, 5, 11, 6, 71, 19, 123, 33, 174, 426, 174, 233, 625, 816, 5695, 3656, 15936, 246960, 24234, 24234, 35151, 140604, 177399, 250982, 1304130, 1304130, 1304130, 1304130, 5532955, 5532955, 58136459, 8525544, 8525544, 58136459, 941988492, 58136459, 941988492

Offset: 1

Author

Andy Niedermaier, May 21 2018

Keywords

• nonn

Comments

Inspired by a test ARML problem from 2018, which asked students to determine a(8).

Examples

			a(2) = a(3) = 0, since no positive squares are achievable.
Some examples:
a(7) = 71 = sqrt(0+1+2*3*4*5*6*7).
a(8) = 19 = sqrt(0+1*2+3+4*5+6*7*8).
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)
		

Links

• Rémy Sigrist, PARI program for A304935

Crossrefs

Upper-bounded by sqrt(A038507).

Programs

Mathematica

sqStrTest[n_] := Module[{bVal, bStr, i, j, iB, mVal, mStr},
  bVal = -1;
  For[i = 0, i < 2^n, i++,
   iB = IntegerDigits[i, 2];
   While[Length[iB] < n, PrependTo[iB, 0]];
   mStr = "0";
   For[j = 1, j <= n, j++,
    mStr = StringJoin[mStr, If[iB[[j]] == 0, "+", "*"], ToString[j]]];
   mVal = ToExpression[mStr];
   If[Sqrt[mVal] == Floor[Sqrt[mVal]],
    If[mVal > bVal, {bVal, bStr} = {mVal, mStr}]
    ]
   ];
  Print[{Sqrt[bVal], bVal, bStr}]]

PARI

See Links section.

Extensions

More terms from Rémy Sigrist, May 22 2018