A299045 -id:A299045 - cp's OEIS frontend

A299107 Probable primes in sequence {s_k(4)}, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5.

5, 19, 71, 3691, 191861, 138907099, 26947261171, 436315574686414344004975231616076636245689199862837798457639364993981991744926792179

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 02 2018

From a problem in A269254. For detailed theory, see [Hone].

Subsequent terms have too many digits to display.

Hugo Pfoertner, Table of n, a(n) for n = 1..14
Gary Greaves and Jose Yip, The coherent rank of a graph with three eigenvalues, arXiv:2406.17395 [math.CO], 2024. See p. 31.
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

A269254, A001834, A294099 (row 4),

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A298675, A298677, A298878, A299045, A299071, A086386.

a(n) = s_{A299100(n)}(4) = A001834(A299100(n)).

A298878 Union_{p prime, n >= 0} {T_p(n)}, where T_m(x) = x*T_{m-1}(x) - T_{m-2}(x), m >= 2, T_0(x) = 2, T_1(x) = x (dilated Chebyshev polynomials of the first kind).

Original entry on oeis.org

-2, -1, 0, 1, 2, 7, 14, 18, 23, 34, 47, 52, 62, 79, 98, 110, 119, 123, 142, 167, 194, 198, 223, 254, 287, 322, 359, 398, 439, 482, 488, 527, 574, 623, 674, 702, 724, 727, 782, 839, 843, 898, 959, 970, 1022, 1087, 1154, 1223, 1294, 1298, 1367, 1442, 1519, 1598

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Jan 27 2018

sign

From a problem in A269254. For detailed theory, see [Hone].

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A008865 (T_2(n)), A299071.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A299045, A299071.

A269251 a(n) = smallest prime in the sequence s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n - 1 (or a(n) = -1 if no such prime exists).

Original entry on oeis.org

-1, -1, 2, 3, 19, 5, 41, 7, 71, 89, 109, 11, 2003, 13, 3121, 239, 271, 17, 729962708557509701, 19, 419, 461, 11593, 23, 599, 11356201, 701, 11546481261621528160662473705515857458665002781273993, 811, 29, 929

Offset: 1

Arkadiusz Wesolowski, Jul 09 2016

sign

For n >= 2, smallest prime of the form (x^y + 1/x^y)/(x + 1/x), where x = (sqrt(n+2) +- sqrt(n-2))/2 and y is an odd positive integer, or -1 if no such prime exists.
If a(34) > 0 then a(34) > 10^1000. - Robert Israel, Feb 06 2018
For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018
Values of n where a(n) might need more than 1000 digits: 34, 52, 123, 254, 275, 285, 322, 371, 401, 413, 437, 460, 508, 518, 535, 540, 629, 643, 653, 691, 723, 724, 753, 797, 837, 843, 876, 881, 898, 913, 960, 970, 981, 986, 987, ... - Jean-François Alcover, Mar 01 2018

C. K. Caldwell, Top Twenty page, Lehmer number
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.
Wikipedia, Lehmer number

Cf. A269252, A269253, A269254.

Cf. A294099, A298675, A298677, A298878, A299045, A299071, A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501.

lst:=[]; for n in [1..31] do if n le 2 then Append(~lst, 0); else a:=1; c:=1; repeat b:=n*a-c; c:=a; a:=b; until IsPrime(a); Append(~lst, a); end if; end for; lst;

f:= proc(n) local a,b,t;
a:= 1; b:= n-1;
do
  if isprime(b) then return b fi;
  t:= n*b-a;
  a:= b;
  b:= t;
od
end proc:
f(1):= -1: f(2):= -1:
map(f, [$1..33]); # Robert Israel, Feb 06 2018

max = 10^1000; a[1] = a[2] = -1; a[n_] := Module[{s}, s[0] = 1; s[1] = n-1; s[k_] := s[k] = n s[k-1] - s[k-2]; For[k = 1, s[k] <= max, k++, If[PrimeQ[s[k]], Return[s[k]]]]] /. Null -> -1; Table[a[n], {n, 1, 33}] (* Jean-François Alcover, Mar 01 2018 *)

If n is prime then a(n+1) = n.

Changed the value for the exceptional case from 0 to -1 for consistency with other sequences. - N. J. A. Sloane, Jan 19 2018

A269252 Define a sequence by s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n - 1. a(n) is the smallest index k such that s(k) is prime, or -1 if no such k exists.

Original entry on oeis.org

-1, -1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 14, 1, 2, 2, 3, 1, 2, 5, 2, 36, 2, 1, 2, 1, 15, -1, 6, 2, 3, 1, 2, 2, 6, 1, 3, 1, 2, 2, 2, 1, 2, 3, 2, -1, 3, 1, 2, 2, 2, 6, 3, 1, 2, 1, 30, 3, 2, 2, 2, 1, 2, 5, 2, 1, 5, 1, 6, 3, 2, 6, 3, 1, 8, 6, 14, 1, 3

Offset: 1

Arkadiusz Wesolowski, Jul 09 2016

sign

For n >= 2, positive integer k yielding the smallest prime of the form (x^y + 1/x^y)/(x + 1/x), where x = (sqrt(n+2) +/- sqrt(n-2))/2 and y = 2*k + 1, or -1 if no such k exists.
Every positive term belongs to A005097.
For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

			Let b(k) be the recursive sequence defined by the initial conditions b(0) = 1, b(1) = 10, and the recursive equation b(k) = 11*b(k-1) - b(k-2). a(11) = 2 because b(2) = 109 is the smallest prime in b(k).
Let c(k) be the recursive sequence defined by the initial conditions c(0) = 1, c(1) = 12, and the recursive equation c(k) = 13*c(k-1) - c(k-2). a(13) = 3 because c(3) = 2003 is the smallest prime in c(k).

C. K. Caldwell, Top Twenty page, Lehmer number
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.
Wikipedia, Lehmer number

Cf. A005097, A269251, A269253, A269254, A299045 (array used to compute this sequence).

Cf. A294099, A298675, A298677, A298878, A299071, A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501.

lst:=[]; for n in [1..85] do if n in [1, 2, 34, 52] then Append(~lst, -1); else a:=1; c:=1; t:=0; repeat b:=n*a-c; c:=a; a:=b; t+:=1; until IsPrime(a); Append(~lst, t); end if; end for; lst;

s[k_, m_] := s[k, m] = Which[k == 0, 1, k == 1, 1 + m, True, m s[k - 1, m] - s[k - 2, m]]; Table[SelectFirst[Range[120], PrimeQ@ Abs@ s[#, -n] &] /. k_ /; MissingQ@ k -> -1, {n, 85}] (* Michael De Vlieger, Feb 03 2018 *)

If n is prime then a(n+1) = 1.

cp's OEIS Frontend

A299107 Probable primes in sequence {s_k(4)}, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5.

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A298878 Union_{p prime, n >= 0} {T_p(n)}, where T_m(x) = x*T_{m-1}(x) - T_{m-2}(x), m >= 2, T_0(x) = 2, T_1(x) = x (dilated Chebyshev polynomials of the first kind).

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A269251 a(n) = smallest prime in the sequence s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n - 1 (or a(n) = -1 if no such prime exists).

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A269252 Define a sequence by s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n - 1. a(n) is the smallest index k such that s(k) is prime, or -1 if no such k exists.

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Magma

Mathematica

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