A337210 -id:A337210 - cp's OEIS frontend

A337211 Numbers k such that b(k) < b(j) for all j < k where b(k) = Min_{sqrt(n) - Sum_{i} sqrt(c_i) > 0 with c_i being unique integers}.

1, 2, 3, 4, 5, 6, 10, 12, 14, 15, 18, 21, 22, 24, 26, 29, 34, 35, 38, 40, 42, 46, 47, 79, 89, 184, 226, 264, 269, 299, 419, 426, 505, 545

Offset: 1

Robert G. Wilson v, Aug 19 2020

This sequence is the original definition of A045880.

			a(1) is 1 since sqrt(1) - sqrt(0)                            = 1.00000 ... ;
a(2) is 2 since sqrt(2) - sqrt(1)                            = 0.41421 ...  which is an improvement over a(0);
a(3) is 3 since sqrt(3) - sqrt(2)                            = 0.31783 ... ;
a(4) is 4 since sqrt(4) - sqrt(3)                            = 0.26794 ... ;
a(5) is 5 since sqrt(5) - sqrt(4)                            = 0.23606 ... ;
a(6) is 6 since sqrt(6) - (sqrt(1) + sqrt(2))                = 0.03527 ... ;
7 is not in the sequence since sqrt(7) -  sqrt(6)            = 0.19626 ... which is not an improvement of a(5);
8 is not in the sequence since sqrt(8) - (sqrt(1) + sqrt(3)) = 0.09637 ... which again is not an improvement over a(5);
9 is not in the sequence since sqrt(9) - sqrt(8)             = 0.17157 ... which is not an improvement of a(5);
a(7) is 10 because sqrt(10) - (sqrt(2) + sqrt(3))            = 0.01601 ... ;
a(8) is 12 because sqrt(12) - (sqrt(1) + sqrt(6))            = 0.01461 ... ; etc.

Sean A. Irvine, Table of n, a(n) for n = 1..34
Sean A. Irvine, Java program (github)

Cf. A045880, A333987, A337210.

a(24) corrected by Jinyuan Wang, Aug 23 2020

Title improved, a(26) corrected and a(31)-a(34) from Sean A. Irvine, Mar 23 2021

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)));

cp's OEIS Frontend

A337211 Numbers k such that b(k) < b(j) for all j < k where b(k) = Min_{sqrt(n) - Sum_{i} sqrt(c_i) > 0 with c_i being unique integers}.

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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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