A098602 -id:A098602 - cp's OEIS frontend

A098890 a(n) = A001652(n)*A046090(n)+1 = A098602(n)+1.

1, 13, 421, 14281, 485113, 16479541, 559819261, 19017375313, 646030941361, 21946034630941, 745519146510613, 25325704946729881, 860328449042305321, 29225841562491651013, 992818284675673829101, 33726595837410418538401, 1145711440187278556476513

Offset: 0

Charlie Marion, Nov 03 2004

			a(3) = 14281 = 99^2+70^2-420.

Colin Barker, Table of n, a(n) for n = 0..653
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A001541, A001542, A001652, A046090, A098602.

Vec(-(x^2-22*x+1)/((x-1)*(x^2-34*x+1)) + O(x^30)) \\ Colin Barker, Jul 09 2015

For n>0, a(n) = A001541(n)^2 + A001542(n)^2 - A098602(n-1).
G.f.: -(x^2-22*x+1) / ((x-1)*(x^2-34*x+1)). - Colin Barker, Jul 09 2015
a(n) = (5/8+1/16*(17+12*sqrt(2))^(-n)*(3-2*sqrt(2)+(3+2*sqrt(2))*(17+12*sqrt(2))^(2*n))). - Colin Barker, Mar 02 2016

Name and a(0) corrected by Charlie Marion, Jul 09 2015

A097480 Positive integers n such that 2n-15 is prime.

Original entry on oeis.org

9, 10, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 34, 37, 38, 41, 43, 44, 47, 49, 52, 56, 58, 59, 61, 62, 64, 71, 73, 76, 77, 82, 83, 86, 89, 91, 94, 97, 98, 103, 104, 106, 107, 113, 119, 121, 122, 124, 127, 128, 133, 136, 139, 142, 143, 146, 148, 149, 154, 161, 163

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 19 2004

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A000040, A098602, A098603.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), this sequence (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+15)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[9,200],PrimeQ[2#-15]&] (* Harvey P. Dale, Apr 04 2021 *)

is(n)=isprime(2*n-15) \\ Charles R Greathouse IV, Jul 12 2016

Half of p+15 where p is a prime greater than 2.

A098605 Positive integers n such that 2n - 17 is prime.

Original entry on oeis.org

10, 11, 12, 14, 15, 17, 18, 20, 23, 24, 27, 29, 30, 32, 35, 38, 39, 42, 44, 45, 48, 50, 53, 57, 59, 60, 62, 63, 65, 72, 74, 77, 78, 83, 84, 87, 90, 92, 95, 98, 99, 104, 105, 107, 108, 114, 120, 122, 123, 125, 128, 129, 134, 137, 140, 143, 144, 147, 149, 150, 155, 162

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 20 2004

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A000040, A098602, A098603.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), this sequence (k=17), A097932 (k=19).

(Prime[Range[2,100]]+17)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)

is(n)=isprime(2*n-17) \\ Charles R Greathouse IV, Jul 12 2016

Half of p+17 where p is a prime greater than 2.

A097363 Positive integers n such that 2n-13 is prime.

Original entry on oeis.org

8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 25, 27, 28, 30, 33, 36, 37, 40, 42, 43, 46, 48, 51, 55, 57, 58, 60, 61, 63, 70, 72, 75, 76, 81, 82, 85, 88, 90, 93, 96, 97, 102, 103, 105, 106, 112, 118, 120, 121, 123, 126, 127, 132, 135, 138, 141, 142, 145, 147, 148, 153, 160, 162

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 18 2004

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A000040, A098602.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), this sequence (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+13)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[8,200],PrimeQ[2#-13]&] (* Harvey P. Dale, Apr 26 2013 *)

Half of p+13 where p is a prime greater than 2.

A195538 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(8).

Original entry on oeis.org

5, 12, 145, 420, 4901, 14280, 166465, 485112, 5654885, 16479540, 192099601, 559819260, 6525731525, 19017375312, 221682772225, 646030941360, 7530688524101, 21946034630940, 255821727047185, 745519146510612, 8690408031080165

Offset: 1

Clark Kimberling, Sep 20 2011

See A195500 for a discussion and references.

Conjecture: a(n) = 35*a(n-2) - 35*a(n-4) + a(n-6) with bisections A098602 and A076218. - R. J. Mathar, Sep 21 2011

Cf. A195500, A195539, A195540.

r = Sqrt[8]; z = 24;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
  p[{r, z}]]  (* A195538, A195539 *)
Sqrt[a^2 + b^2] (* A195540 *)
(* Peter J. C. Moses, Sep 02 2011 *)

cp's OEIS Frontend

A098890 a(n) = A001652(n)*A046090(n)+1 = A098602(n)+1.

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A097480 Positive integers n such that 2n-15 is prime.

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Mathematica

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A098605 Positive integers n such that 2n - 17 is prime.

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A097363 Positive integers n such that 2n-13 is prime.

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A195538 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(8).

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Mathematica