A004273 -id:A004273 - cp's OEIS frontend

A166263 a(j) = maximum value of n for each distinct increasing value of (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic residues of prime(n)) / prime(n) for each j.

348511, 38, 155, 389, 778, 1296, 1828, 2321, 3683, 3935, 4078, 6184, 8783, 9013, 9880, 15182, 12449, 19828, 18884, 14593, 22316, 25738, 26064, 26670, 31953, 33332, 45025, 35788, 37881, 50299, 39562, 49598, 77850, 56777, 53024, 70443, 71992

Offset: 1

Christopher Hunt Gribble, Oct 10 2009

nonn

a(1) appears to increase indefinitely, so the static sequence starts from a(2).
The value of a(1) is the index of the largest prime p < 5*10^6 for which Sum of the quadratic non-residues of p = Sum of the quadratic residues of p.
The table below shows for each value of a(j) the corresponding values of p(a(j)) and (Sum of the quadratic non-residues of p(a(j)) - Sum of the quadratic residues of p(a(j))) / p(a(j)):
.
   j      a(j)    prime(a(j))   (SQN-SQR)/prime(a(j))
  --    ------    -----------   ---------------------
  348511    4999961          0
      38        163          1
     155        907          3
     389       2683          5
     778       5923          7
    1296      10627          9
    1828      15667         11
    2321      20563         13
    3683      34483         15
    3935      37123         17
    4078      38707         19
    6184      61483         21
    8783      90787         23
    9013      93307         25
    9880     103387         27
   15182     166147         29
   12449     133387         31
   19828     222643         33
   18884     210907         35
   14593     158923         37
   22316     253507         39
   25738     296587         41
   26064     300787         43
   26670     308323         45
   31953     375523         47
   33332     393187         49
   45025     546067         51
   35788     425107         53
   37881     452083         55
   50299     615883         57
   39562     474307         59
   49598     606643         61
   77850     991027         63
   56777     703123         65
   53024     652723         67
   70443     888427         69
   71992     909547         71
   70328     886867         73
   72479     916507         75

Christopher Hunt Gribble, Table of n, a(n) for n = 1..1973.

Cf. A165951, A165974, A004273.

A165520 Antidiagonal writing from three rows trio A165351,A165355,A165367 (first,second and third trisections of A026741).

Original entry on oeis.org

0, 1, 3, 1, 2, 3, 5, 7, 9, 4, 5, 6, 11, 13, 15, 7, 8, 9, 17, 19, 21, 10, 11, 12, 23, 25, 27, 13, 14, 15, 29, 31, 33

Offset: 1

Paul Curtz, Sep 21 2009

(6n+3)-th term is 6n+3=A016945. 6n-th term is 3n=3*n=A008585.

Mix (A004273(3n),A004273(3n+1),A004273(3n+2)), (A000027(3n),A000027(3n+1),A000027(3n+2)).

A286016 Signed continued fraction expansion with all signs negative of tanh(1).

Original entry on oeis.org

1, 5, 2, 2, 2, 2, 9, 2, 2, 2, 2, 2, 2, 2, 2, 13, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 17, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 21, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 25, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Kutlwano Loeto, Apr 30 2017

For any given sequence of signs (e_1, e_2, ..., e_n, ...) one may define the signed continued fraction expansion of a real number x by using floor or ceiling in the step i according to e_i = +1 or e_i = -1. In the present case for the sequence (-1, -1, -1, -1, ...) consisting of only negative signs the ceiling is taken at each step, and the formulas with x_0 = x are a_n = ceiling(x_n) and x_{n+1} = 1/(a_n - x_n). See chapter 1 and 2 of the book by Perron.

			a(2) = 5, a(3) = a(4) = a(5) = a(6) = 2, a(7) = 9, etc. These numbers are obtained from the partial quotients xj as follows:
x2 =  (1 +  e^2)/( 2 + 0e^2) ~4.17 so that a(2)=ceiling(x2)=5;
x3 =  (2 + 0e^2)/( 9 - e^2)  ~1.21 so that a(3)=ceiling(x3)=2;
x4 =  (9 -  e^2)/(16 - 2e^2) ~1.31 so that a(4)=ceiling(x4)=2;
x5 = (16 - 2e^2)/(23 - 3e^2) ~1.46 so that a(5)=ceiling(x5)=2;
x6 = (23 - 3e^2)/(30 - 4e^2) ~1.87 so that a(6)=ceiling(x6)=2;
x7 = (30 - 4e^2)/(37 - 5e^2) ~8.11 so that a(7)=ceiling(x7)=9.
The pairs of integers appearing in the xj's are obtained as the principal or as every other of the non-principal approximating fractions of e^2 in the sense of the A. Hurwitz reference.

Adolf Hurwitz, Über die Kettenbrüche, deren Teilnenner arithmetische Reihen bilden, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, Jahrg. 41 (1896), Jubelband 2, 34-64. Reprinted in Hurwitz's Mathematische Werke.
Oskar Perron, Die Lehre von den Kettenbrüchen, Teubner, Leipzig, 1930.

Cf. A004273 (continued fraction of tanh(1)), A280135, A280136.

x:=(exp(1)-exp(-1))/(exp(1)+exp(-1)):b:=ceil(x): x1:=1/(b-x):L:=[b]:
for k from 0 to 40 do:
b1:=ceil(x1): x1:=1/(b1-x1): L:=[op(L),b1]: od: print(L);

Using an obvious condensed notation we get for the sequence 1, 5, 2^(4), 9, 2^(8), 13, 2^(12), 17, 2^(16), 21, 2^(20), ... where 2^(m) means m copies of 2.

More terms from Jinyuan Wang, Jul 02 2022

cp's OEIS Frontend

A166263 a(j) = maximum value of n for each distinct increasing value of (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic residues of prime(n)) / prime(n) for each j.

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A165520 Antidiagonal writing from three rows trio A165351,A165355,A165367 (first,second and third trisections of A026741).

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A286016 Signed continued fraction expansion with all signs negative of tanh(1).

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