A117522 -id:A117522 - 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.

A331478 Irregular triangle T(n,k) = n - (s - k + 1)^2 for 1 <= k <= s, with s = floor(sqrt(n)).

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 2, 5, 3, 6, 4, 7, 0, 5, 8, 1, 6, 9, 2, 7, 10, 3, 8, 11, 4, 9, 12, 5, 10, 13, 6, 11, 14, 0, 7, 12, 15, 1, 8, 13, 16, 2, 9, 14, 17, 3, 10, 15, 18, 4, 11, 16, 19, 5, 12, 17, 20, 6, 13, 18, 21, 7, 14, 19, 22, 8, 15, 20, 23, 0, 9, 16, 21, 24, 1

Offset: 1

Michael De Vlieger, Jan 17 2020

Row n begins with n - floor(sqrt(n)).
Zero appears in row n for n that are perfect squares. Let r = sqrt(n). For perfect square n, there exists a partition of n that consists of a run of r parts that are each r themselves; e.g., for n = 4, we have {2, 2}, for n = 9, we have {3, 3, 3}. It is clear through the Ferrers diagram of these partitions that they are equivalent to their Durfee square, thus n - s^2 = 0.
Since the partitions of any n contain Durfee squares in the range of 1 <= s <= floor(sqrt(n)) (with perfect square n also including k = 0), the distinct Durfee square excesses must be the differences n - s^2 for 1 <= s <= floor(sqrt(n)).
We borrow the term "square excess" from A053186(n), which is simply the difference n - floor(sqrt(n)).
Row n of this sequence contains distinct Durfee square excesses among all integer partitions of n (see example below).

			Table begins:
   1:  0;
   2:  1;
   3:  2;
   4:  0,  3;
   5:  1,  4;
   6:  2,  5;
   7:  3,  6;
   8:  4,  7;
   9:  0,  5,  8;
  10:  1,  6,  9;
  11:  2,  7, 10;
  12:  3,  8, 11;
  13:  4,  9, 12;
  14:  5, 10, 13;
  15:  6, 11, 14;
  16:  0,  7, 12, 15;
  ...
For n = 4, the partitions are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}. The partition {2, 2} has Durfee square s = 2; for all partitions except {2, 2}, we have Durfee square with s = 1. Therefore we have two unique solutions to n - s^2 for n = 4, i.e., {0, 3}, so row 4 contains these values.

Michael De Vlieger, Table of n, a(n) for n = 1..10125 (rows 1 <= n <= 625, flattened)
Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A000196, A022554, A053186, A117522.

Array[# - Reverse@ Range[Sqrt@ #]^2 &, 625] // Flatten

Let s = floor(sqrt(n));
T(n,1) = A053186(n) = n - s;
T(n,k) = T(n,1) + partial sums of 2(s - k + 1) + 1 for 2 <= k <= s + 1.
A000196(n) = Length of row n.
A022554(n) = Sum of row n.
Last term in row n = T(n, A000196(n)) = n - 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

Formula

A331478 Irregular triangle T(n,k) = n - (s - k + 1)^2 for 1 <= k <= s, with s = floor(sqrt(n)).

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