This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(7) = 8 since A364844(7)/A364845(7) = 7327237/2332332 = 3.1415926206046... matches the first 8 digits of Pi: 3.1415926535897...
n fraction approximated value - ------------------- ------------------ 1 3/1 3 2 11/4 2.75 3 878/323 2.7182662538699... 4 2552/939 2.7177848775292... 5 38983/14341 2.7182902168607... 6 167761/61716 2.7182740294251... 7 4407044/1621261 2.7182816338640... 8 24988942/9192919 2.7182815382143... 9 882646288/324707423 2.7182818299783... ...
a[1]=3; a[n_]:=Module[{minim = Infinity}, h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; lh = Length[h]; For[i = 1, i <= lh, i++, k = Select[Range[Floor[Part[h, i]/E], Ceiling[Part[h, i]/E]], PalindromeQ]; lk = Length[k]; For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - E]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; hmin = Part[h, i]]]]; hmin]; Array[a,9]
\\ See PARI program in Links
n fraction approximated value - ------------------- ------------------ 1 3 3 2 22/7 3.1428571428571... 3 474/151 3.1390728476821... 4 1551/494 3.1396761133603... 5 36163/11511 3.1416036834332... 6 292292/93039 3.1416072829673... 7 7327237/2332332 3.1415926206046... 8 31311313/9966699 3.1415931192464... ...
nmax = 8; a = {1}; hmin = kmin = 0; For[n = 2, n <= nmax, n++, minim = Infinity; h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; k = Select[Range[10^(n - 2), 10^n - 1], PalindromeQ]; lh = Length[h]; lk = Length[k]; For[i = 1, i <= lh, i++, For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - Pi]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; kmin = Part[k, j]]]]; AppendTo[a, kmin]]; a
n (h/k)^4 approximated value - ------------------- ------------------ 1 (4/3)^4 3.1604938271604... 2 (97/73)^4 3.1174212867620... 3 (888/667)^4 3.1415829223858... 4 (9551/7174)^4 3.1415927852873... 5 (13549/10177)^4 3.1415926560044... ...
nmax = 3; a = {}; hmin = kmin = 0; For[n = 1, n <= nmax, n++, minim = Infinity; For[h = 10^(n-1), h <10^n, h++, For[k = 1, k < 10^n/Pi^(1/4), k++, If[(dist = Abs[h^4/k^4-Pi]) < minim && GCD[h,k]==1, minim = dist; hmin=h; kmin = k]]]; AppendTo[a, hmin]]; a
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