A013645 -id:A013645 - cp's OEIS frontend

A020640 Successive record periods of continued fraction for sqrt(k) (period of continued fraction for sqrt(A013645(n+1))).

1, 2, 4, 5, 6, 8, 10, 12, 16, 18, 20, 22, 26, 34, 37, 40, 42, 44, 48, 52, 54, 60, 62, 64, 70, 76, 79, 88, 94, 96, 102, 104, 108, 114, 118, 122, 130, 136, 152, 156, 158, 170, 172, 174, 194, 196, 202, 207, 210, 217, 228, 234, 238, 239, 248, 262, 268, 280, 281

Offset: 1

David W. Wilson

Patrick McKinley, Table of n, a(n) for n=1..512

Cf. A003285, A013645.

per[n_] := ContinuedFraction[Sqrt[n]][[2]] // Length; record = 0; t = Reap[For[n = 1, n < 2*10^4, n++, If[ !IntegerQ[Sqrt[n]], p = per[n]; If[p > record, record = p; Print[{n, p}]; Sow[{n, p}]]]]][[2, 1]]; A020640 = t[[All, 2]] (* Jean-François Alcover, Dec 27 2012 *)
DeleteDuplicates[Table[If[IntegerQ[Sqrt[n]],Nothing,Length[ContinuedFraction[Sqrt[n]][[2]]]],{n,20000}],GreaterEqual] (* Harvey P. Dale, Dec 28 2023 *)

a(n) = A003285(A013645(n+1)). - Pontus von Brömssen, Nov 24 2024

A130272 Primes p for which the period of the continued fraction of sqrt(p) increases.

Original entry on oeis.org

2, 3, 7, 13, 19, 31, 43, 61, 103, 109, 139, 151, 181, 211, 331, 421, 541, 571, 631, 751, 919, 1291, 1381, 1549, 1579, 1621, 1759, 1831, 2011, 2311, 2671, 3019, 3469, 3691, 3931, 4909, 4951, 4999, 5119, 6211, 6451, 6679, 8269, 8719, 8779, 8941, 9739, 9949

Offset: 1

T. D. Noe, May 19 2007

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..344 (terms n = 1..200 from T. D. Noe)

Cf. A003285, A013645.

mx=0; n=0; t=Table[n++; While[p=Prime[n]; len=Length[Last[ContinuedFraction[Sqrt[p]]]]; len<=mx, n++ ]; mx=len; p, {50}]

Where records occur in A054269.

A374232 Indices of records in A028832.

Original entry on oeis.org

1, 2, 3, 14, 19, 46, 94, 151, 211, 379, 526, 694, 919, 1324, 1759, 2011, 2326, 3691, 4174, 5086, 6451, 7606, 8254, 10294, 10651, 13126, 17599, 18979, 19231, 21319, 30319, 31606, 32971, 34654, 42379, 46006, 48799, 58774, 76651, 78094, 82471, 85999, 90931, 101599

Offset: 1

Amiram Eldar, Jul 01 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A013645, A028832.

s[n_] := If[IntegerQ@ Sqrt[n], 0, Length @ DeleteDuplicates[ContinuedFraction[Sqrt[n]][[2]]]];
seq[kmax_] := Module[{smax = -1, s1, sq = {}}, Do[If[(s1 = s[k]) > smax, smax = s1; AppendTo[sq, k]], {k, 1, kmax}]; sq]; seq[10^5]

A233423 Values of n at which the period of the continued fraction for sqrt(n) is nondecreasing.

Original entry on oeis.org

1, 2, 3, 6, 7, 13, 19, 21, 22, 31, 43, 46, 76, 94, 124, 133, 139, 151, 166, 211, 214, 244, 301, 309, 331, 421, 526, 571, 604, 631, 751, 886, 919, 991, 1279, 1291, 1324, 1366, 1516, 1621, 1726, 2011, 2311, 2566, 2671, 3004, 3019, 3259, 3334, 3691, 3931, 4174

Offset: 1

David Spies, Dec 09 2013

nonn

Superset of A013645.

t = {1}; mx = 0; n = 1; While[Length[t] < 60, n++; If[! IntegerQ[Sqrt[n]], len = Length[ContinuedFraction[Sqrt[n]][[2]]]; If[len >= mx, AppendTo[t, n]; mx = len]]]; t (* T. D. Noe, Dec 10 2013 *)

Extended by T. D. Noe, Dec 10 2013

cp's OEIS Frontend

A020640 Successive record periods of continued fraction for sqrt(k) (period of continued fraction for sqrt(A013645(n+1))).

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A130272 Primes p for which the period of the continued fraction of sqrt(p) increases.

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Mathematica

Formula

A374232 Indices of records in A028832.

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A233423 Values of n at which the period of the continued fraction for sqrt(n) is nondecreasing.

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