A337211 -id:A337211 - cp's OEIS frontend

A337210 Irregular triangle read by rows in which row n has the least number of integers such that the sum of the square root of those integers is the best approximation to and less than the square root of n.

0, 1, 2, 3, 4, 1, 2, 6, 1, 3, 8, 2, 3, 1, 5, 1, 6, 12, 3, 4, 2, 6, 3, 5, 2, 7, 4, 5, 1, 11, 1, 12, 2, 10, 5, 6, 1, 14, 3, 10, 1, 3, 5, 6, 7, 1, 3, 6, 2, 15, 2, 3, 5, 7, 8, 1, 3, 8, 2, 4, 5, 4, 14, 8, 9, 6, 12, 2, 21, 1, 5, 8, 9, 10, 4, 18, 6, 15, 1, 5, 10, 10

Offset: 1

Robert G. Wilson v, Aug 19 2020

All approximations are less than or equal to one. An approximation sqrt(n) - sqrt(n-1) < 1 for all n > 1.
Often integers of the form 4n-2 have as their best approximation just the two consecutive integers {n-1, n}.
Those that are not: 20, 21, 25, 27, 30, 31, 36, 37, 38, 40, 42, 44, 45, 46, 47, 48, 49, 52, ... .
Sometimes two approximations are equal, i.e.; for n = 39, sqrt(2) + sqrt(4) + sqrt(8) is the same as sqrt(4) + sqrt(18). In this sequence the simplest form is used, i.e.; {4, 18}.

			For row 1, just the sqrt(0) < sqrt(1);
for row 2, just the sqrt(1) < sqrt(2);
for row 3, just the sqrt(2) < sqrt(3);
for row 4, just the sqrt(3) < sqrt(4);
for row 5, just the sqrt(4) < sqrt(5);
for row 6, sqrt(1) + sqrt(2) < sqrt(6);
for row 7, just the sqrt(6) < sqrt(7);
for row 8, sqrt(1) + sqrt(3) < sqrt(8);
for row 9, just the sqrt(8) < sqrt(9);
for row 10, sqrt(2) + sqrt(3) is the best approximation;
for row 11, sqrt(1) + sqrt(5) < sqrt(11);
for row 12, sqrt(1) + sqrt(6) < sqrt(12);
for row 27, sqrt(1) + sqrt(3) + sqrt(6) is the best approximation;
for row 63, 2*sqrt(3) + 2*sqrt(5) is the best approximation and appears as the integers {12, 20};
for row 107, sqrt(3) + sqrt(6) + sqrt(9) + sqrt(10) is the best approximation;
for row 165, sqrt(1) + 2*sqrt(2) + 2*sqrt(3) + sqrt(5) + sqrt(11) is the best approximation and appears as the integers {1, 5, 8, 11, 12};
for row 218, sqrt(1) + sqrt(3) + sqrt(5) + sqrt(6) + sqrt(13) + sqrt(14) is the best approximation; etc.
Triangle begins:
0;
1;
2;
3;
4;
1, 2;
6;
1, 3;
8;
2, 3;
...

Inspired by A045880.

Cf. A337211.

y[x_] := Block[{lst = {x - 1}, min = Sqrt[x] - Sqrt[x - 1], rad = 1, sx = Sqrt[x]},
  If[x > 5, lim = (sx - 1)^2;
   Do[diff = sx - (Sqrt[a] + Sqrt[b]);
    If[diff < min && diff > 0, min = diff; lst = {b, a}; rad = 2],
    {a, 2, lim}, {b, 1, a - 1}]];
  If[x > 17, lim = (sx - Sum[Sqrt[z], {z, 2}])^2;
   Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c]);
    If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff; lst = {c, b, a}; rad = 3],
    {a, 3, lim}, {b, 2, a - 1}, {c, 1, b - 1}]];
  If[x > 37, lim = (sx - Sum[Sqrt[z], {z, 3}])^2;
   Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d]);
    If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff; lst = {d, c, b, a};
     rad = 4],
    {a, 4, lim}, {b, 3, a - 1}, {c, 2, b - 1}, {d, 1, c - 1}]];
  If[x > 71, lim = (sx - Sum[Sqrt[z], {z, 4}])^2;
   Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e]);
    If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff; lst = {e, d, c, b, a};
     rad = 5],
    {a, 5, lim}, {b, 4, a - 1}, {c, 3, b - 1}, {d, 2, c - 1}, {e, 1,
     d - 1}]];
  If[x > 117, lim = (sx - Sum[Sqrt[z], {z, 5}])^2;
   Do[diff =
     sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f]);
     If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff; lst = {f, e, d, c, b, a};
     rad = 6],
    {a, 6, lim}, {b, 5, a - 1}, {c, 4, b - 1}, {d, 3, c - 1}, {e, 2,
     d - 1}, {f, 1, e - 1}]];
  If[x > 181, lim = (sx - Sum[Sqrt[z], {z, 6}])^2;
   Do[diff =
     sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f] +
         Sqrt[g]); If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff;
     lst = {g, f, e, d, c, b, a}; rad = 7],
    {a, 7, lim}, {b, 6, a - 1}, {c, 5, b - 1}, {d, 4, c - 1}, {e, 3,
     d - 1}, {f, 2, e - 1}, {g, 1, f - 1}]];
  If[x > 265, lim = (sx - Sum[Sqrt[z], {z, 7}])^2;
   Do[diff =
     sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f] +
         Sqrt[g] + Sqrt[h]); If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff;
     lst = {g, f, e, d, c, b, a}; rad = 8],
    {a, 8, lim}, {b, 7, a - 1}, {c, 6, b - 1}, {d, 5, c - 1}, {e, 4,
     d - 1}, {f, 3, e - 1}, {g, 2, f - 1}, {h, 1, g - 1}]];
 lst];
 Array[ y, 50] // Flatten

s = sum(sqrt(i)) for carefully chosen integers i less than n such that s < n yet is the best approximation to n.

A333987 Integers m for which b(m) < b(m-1) where b(k) = Min_{sqrt(k) - sqrt(x) - sqrt(y) > 0 with x, y distinct integers}.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 14, 15, 18, 21, 22, 24, 26, 30, 33, 34, 35, 38, 40, 42, 46, 50, 54, 58, 60, 62, 63, 65, 66, 70, 74, 78, 82, 84, 85, 86, 90, 94, 98, 99, 102, 106, 110, 112, 114, 118, 122, 126, 130, 133, 134, 138, 142, 143, 144, 146, 150, 154, 158, 161

Offset: 1

Jinyuan Wang and Robert G. Wilson v, Sep 04 2020

b(a(n)) is a closer approximation than b(a(n-1)), where b(k) is the "best approximation" to k using only two radicals as defined in A337210.
Except for the first five terms, all terms present require two positive radicals.
Numbers of the form 4k - 2 for k > 0 are always in the sequence with arguments of their two radicals being k - 1 and k.
4, 12, 24, 40, 60, 84, 112, 144, 180, ... are terms == 0 (mod 4); 1, 5, 21, 33, 65, 85, 133, 161, 225, ... are terms == 1 (mod 4); 3, 15, 35, 63, 99, 143, 195, 255, 323, ... are terms == 3 (mod 4).

Cf. A045880, A337210, A337211.

y[x_] := Block[{lst = {x - 1}, min = Sqrt[x] - Sqrt[x - 1], rad = 1, sx = Sqrt[x]}, If[x > 5, a = 2; lim = (sx - 1)^2; While[a <= lim, b = 1; While[b < a, diff = sx - (Sqrt[a] + Sqrt[b]); If[ diff < 0, Break[]]; If[diff < min && diff > 0, rad = 2; min = diff; lst = {b, a}]; b++]; a++]]; min]; k = 1; min = Infinity; lst = {}; While[k < 171, a = y@k; If[a < min, min = a; AppendTo[lst, k]]; k++]; lst

b(k) = {my(m=s=sqrt(k), t); for(x=1, k\4, if((t=(t=s-sqrt(x))-sqrt(floor(t^2))) < m && t > 10^-20, m=t)); m; }
lista(nn) = my(r=1); for(k=1, 4, print1(k, ", ")); for(k=1, nn, if(b(k) < r, print1(k, ", "); r=b(k)));

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A045880 Integers whose square root is "well approximated" by other square roots of integers.

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A337210 Irregular triangle read by rows in which row n has the least number of integers such that the sum of the square root of those integers is the best approximation to and less than the square root of n.

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A333987 Integers m for which b(m) < b(m-1) where b(k) = Min_{sqrt(k) - sqrt(x) - sqrt(y) > 0 with x, y distinct integers}.

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