A088165 -id:A088165 - cp's OEIS frontend

A117522 Numbers k such that L(2*k + 1) is prime, where L(m) is a Lucas number.

2, 3, 5, 6, 8, 9, 15, 18, 20, 23, 26, 30, 35, 39, 56, 156, 176, 251, 306, 308, 431, 548, 680, 2393, 2396, 2925, 3870, 4233, 5345, 6125, 6981, 7224, 9734, 17724, 18389, 22253, 25584, 28001, 40835, 44924, 47411, 70028, 74045, 79760, 91544, 96600, 101333, 172146, 193716, 221804, 266138, 287109, 308393, 315590, 318875, 325910, 346073, 450828, 525924

Offset: 1

Parthasarathy Nambi, Apr 26 2006

For n = 24..43, we can only claim that L(2*a(n) + 1) is a probable prime. Sequence arises in a study of A269254; for detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

			If k = 56, then L(2*k + 1) is a prime with twenty-four digits.

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

Cf. A000032, A001606, A269251, A269252, A269253, A269254.

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

Values beyond 680 from L. Edson Jeffery, et al., Feb 02 2018
a(44)-a(56) from Robert Price, Jun 12 2025
a(57)-a(59) (using data in A001606) from Alois P. Heinz, Jun 12 2025

A269253 Smallest prime in the sequence s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n + 1 (or -1 if no such prime exists).

Original entry on oeis.org

2, 3, 11, 5, 29, 7, -1, 71, 89, 11, 131, 13, 181, -1, 239, 17, 5167, 19, 379, 419, 461, 23, -1, 599, 251894449, 701, 20357, 29, 25171, 31, 991, 36002209323169, 47468744103199, -1, 1259, 37, 2625505273, 1481, 1559, 41, 1721, 43, 150103799, 1979, 2069, 47, -1, 2351, 287762399

Offset: 1

Arkadiusz Wesolowski, Jul 09 2016

sign

For n >= 3, 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.
When n=7, the sequence {s(k)} is A033890, which is Fibonacci(4i+2), and since x|y <=> F_x|F_y, and 2i+1|4i+2, A033890 is never prime, and so a(7) = -1. What is the proof for the other entries that are -1? Answer: See the Comments in A269254. - N. J. A. Sloane, Oct 22 2017
For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

Hans Havermann, Table of n, a(n) for n = 1..300
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. A117522, A269251, A269252, A269254.

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

lst:=[]; for n in [1..49] do if n gt 2 and IsSquare(n+2) then Append(~lst, -1); else a:=n+1; c:=1; if IsPrime(a) then Append(~lst, a); else repeat b:=n*a-c; c:=a; a:=b; until IsPrime(a); Append(~lst, a); end if; end if; end for; lst;

terms = 172;
kmax = 120;
a[n_] := Module[{s, k}, s[k_] := s[k] = n s[k-1] - s[k-2]; s[0] = 1; s[1] = n+1; For[k = 1, k <= kmax, k++, If[PrimeQ[s[k]], Return[s[k]]]]];
Array[a, terms] /. Null -> -1 (* Jean-François Alcover, Aug 30 2018 *)

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

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, -1, -1, 1, -2, 1, 1, 1, -3, 5, -1, 0, 1, -4, 11, -13, 1, -1, 1, -5, 19, -41, 34, -1, 1, 1, -6, 29, -91, 153, -89, 1, 0, 1, -7, 41, -169, 436, -571, 233, -1, -1, 1, -8, 55, -281, 985, -2089, 2131, -610, 1, 1, 1, -9, 71, -433, 1926, -5741, 10009, -7953, 1597, -1, 0

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 01 2018

This array is used to compute A269252: A269252(n) = least k such that |A(n,k)| is a prime, or -1 if no such k exists.
For detailed theory, see [Hone].
The array can be extended to k<0 with A(n, k) = -A(n, -k-1) for all k in Z. - Michael Somos, Jun 19 2023

			Array begins:
1   0  -1     1     0      -1       1         0        -1           1
1  -1   1    -1     1      -1       1        -1         1          -1
1  -2   5   -13    34     -89     233      -610      1597       -4181
1  -3  11   -41   153    -571    2131     -7953     29681     -110771
1  -4  19   -91   436   -2089   10009    -47956    229771    -1100899
1  -5  29  -169   985   -5741   33461   -195025   1136689    -6625109
1  -6  41  -281  1926  -13201   90481   -620166   4250681   -29134601
1  -7  55  -433  3409  -26839  211303  -1663585  13097377  -103115431
1  -8  71  -631  5608  -49841  442961  -3936808  34988311  -310957991
1  -9  89  -881  8721  -86329  854569  -8459361  83739041  -828931049

Robert Price, Table of n, a(n) for n = 1..5050
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269251, A269252, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
Cf. A094954 (unsigned version of this array, but missing the first row).
Cf. Rows: A057078, A033999, A099496, A079935 (or A001835), A004253, A001653, A049685, A070997, A070998, A138288 (or A072256), ...
Cf. Columns: A000012, A001477 (A000027), A110331 (A165900), A123972, A192398, ...

(* Array: *)
Grid[Table[LinearRecurrence[{-n, -1}, {1, 1 - n}, 10], {n, 10}]]
(*Array antidiagonals flattened (gives this sequence):*)
A299045[n_, k_] := Sum[(-1)^(Floor[j/2]) Binomial[k - Floor[(j + 1)/2], Floor[j/2]] (-n)^(k - j), {j, 0, k}]; Flatten[Table[A299045[n - k, k], {n, 11}, {k, 0, n - 1}]]

{A(n, k) = sum(j=0, k, (-1)^(j\2)*binomial(k-(j+1)\2, j\2)*(-n)^(k-j))}; /* Michael Somos, Jun 19 2023 */

G.f. for row n: (1 + x)/(1 + n*x + x^2), n >= 1.

A(n, k) = B(-n, k) where B = A294099. - Michael Somos, Jun 19 2023

A299071 Union_{odd primes p, n >= 3} {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

18, 52, 110, 123, 198, 488, 702, 724, 843, 970, 1298, 1692, 2158, 2525, 3330, 4048, 4862, 5778, 6726, 6802, 7940, 9198, 10084, 10582, 13752, 15550, 17498, 19602, 21868, 24302, 26910, 29698, 30248, 32672, 35838, 39603, 42770, 46548, 50542

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 01 2018

nonn

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

Sequence avoids numbers of the form T_p(T_2(j)).

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)), A298878 (T_p(n), p prime).

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

maxT = 55000; maxn = 12;
T[0][] = 2; T[1][x] = x;
T[m_][x_] := T[m][x] = x T[m-1][x] - T[m-2][x];
TT = Table[T[p][n], {p, Prime[Range[2, maxn]]}, {n, 3, Prime[maxn]}] // Flatten // Union // Select[#, # <= maxT&]&;
avoid = Table[T[p][T[2][n]], {p, Prime[Range[2, maxn]]}, {n, 3, Prime[maxn] }] // Flatten // Union // Select[#, # <= maxT&]&;
Complement[TT, avoid] (* Jean-François Alcover, Nov 03 2018 *)

A299100 Indices k such that s_k(4) is a (probable) prime, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 18, 146, 216, 293, 704, 1143, 1530, 1593, 2924, 7163, 9176, 9489, 11531, 39543, 50423, 60720, 62868, 69993, 69995, 88103, 88163, 104606, 164441, 178551

Offset: 1

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

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

a(31) > 2*10^5. - Robert Price, May 29 2020

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

Cf. A269254. Indices of primes in A001834, A294099 (row 4).

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

s[k_, m_] := s[k, m] = Which[k == 0, 1, k == 1, 1 + m, True, m s[k - 1, m] - s[k - 2, m]]; Select[Range@ 2000, PrimeQ@ Abs@ s[#, 4] &] (* Michael De Vlieger, Feb 03 2018 *)

a(24)-a(30) from Robert Price, May 29 2020

A299101 Indices of (probable) primes in A030221.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 15, 18, 23, 53, 114, 194, 564, 575, 585, 2594, 3143, 4578, 4970, 9261, 11508, 13298, 30018, 54993, 198476

Offset: 1

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

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

a(25) > 2*10^5. - Robert Price, Jul 03 2020

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

Cf. A269254, A030221, A294099 (row 5).

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

-1 + Position[Nest[Append[#, 5 #[[-1]] - #[[-2]] ] &, {1, 6}, 2600], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Jul 03 2020 *)

A299109(n) = A030221(a(n)). - R. J. Mathar, Jul 22 2022

a(24) from Robert Price, Jul 03 2020

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.

Original entry on oeis.org

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.

A066885 a(n) = (prime(n)^2 + 1)/2.

Original entry on oeis.org

5, 13, 25, 61, 85, 145, 181, 265, 421, 481, 685, 841, 925, 1105, 1405, 1741, 1861, 2245, 2521, 2665, 3121, 3445, 3961, 4705, 5101, 5305, 5725, 5941, 6385, 8065, 8581, 9385, 9661, 11101, 11401, 12325, 13285, 13945, 14965, 16021, 16381, 18241, 18625

Offset: 2

Enoch Haga, Jan 22 2002

a(n) is the average of the numbers from 1 to prime(n)^2. It's also the average of the primes in a prime(n) X prime(n) example of Haga's conjecture (see link below).
If a(n) is a square c^2, then prime(n) is an NSW prime (A088165) and a prime RMS number (A140480). - Ctibor O. Zizka, Aug 26 2008
The sequence starts with a(2) = (3^2 + 1)/2 = 5 since a(1) would be (2^2 + 1)/2 = 5/2. - Michael B. Porter, Dec 14 2009

Harry J. Smith, Table of n, a(n) for n = 2..1000
Carlos Rivera, Conjecture 26. The Calendar-like square Conjecture, The Prime Puzzles and Problems Connection.

Cf. A066883, A066886.
Cf. A088165, A140480.
Cf. A084921.
Partial sums of A124434.

[(NthPrime(n)^2+1)/2 : n in [2..50]]; // Wesley Ivan Hurt, Jun 23 2015

A066885:=n->(ithprime(n)^2+1)/2: seq(A066885(n), n=2..50); # Wesley Ivan Hurt, Jun 23 2015

a[n_] := (Prime[n]^2+1)/2; Table[a[n], {n, 2, 50}]

A066885(n) = (prime(n)^2+1)/2 \\ Michael B. Porter, Dec 14 2009

{ for (n=2, 1000, write("b066885.txt", n, " ", (prime(n)^2 + 1)/2) ) } \\ Harry J. Smith, Apr 04 2010

a(n) = 1 + A084921(n). - R. J. Mathar, Sep 30 2011
a(n) mod 4 = 1. - Altug Alkan, Apr 08 2016
Product_{n>=2} (1 - 1/a(n)) = 2/3. - Amiram Eldar, Jun 03 2022

Edited by Dean Hickerson, Jun 08 2002

A065876 a(n) is the smallest m > n such that n^2 + 1 divides m^2 + 1.

Original entry on oeis.org

1, 3, 3, 7, 13, 21, 31, 43, 18, 73, 91, 111, 17, 47, 183, 211, 241, 133, 57, 343, 381, 47, 172, 83, 553, 601, 651, 173, 342, 813, 242, 265, 132, 403, 411, 1191, 1261, 237, 327, 1483, 1561, 1641, 748, 857, 850, 1981, 684, 463, 413, 2353, 255, 2551, 593, 1177, 2863, 123, 3081, 307, 1288, 3423

Offset: 0

Benoit Cloitre, Dec 07 2001

nonn

a(n) exists because n^2 + 1 divides (n^2 - n + 1)^2 + 1. The set of n such a(n) = n^2 - n + 1 is S = (2, 3, 4, 5, 6, 7, 9, 11, 14, 15, ...).
a(n) = n^2 - n + 1 whenever n^2 + 1 is prime or twice a prime. Up to n=1000, the only other n for which a(n) = n^2 - n + 1 are 7, 41 and 239. Is it a coincidence that these are NSW primes (A088165)? - Franklin T. Adams-Watters, Oct 17 2006
It appears that the density of even numbers in this sequence approaches a limit near 1/4. It appears that the density of even values for indices where a(n) != n^2 - n + 1 is approaching a number near 1/4 and based on the previous comment the density of n for which a(n) = n^2 - n + 1 is almost certainly 0. - Franklin T. Adams-Watters, Oct 17 2006

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..1000

Cf. A002731, A005574, A071557, A088165.

Do[k = 1; While[m = (k^2 + 1)/(n^2 + 1); m < 2 || !IntegerQ[m], k++ ]; Print[k], {n, 1, 40 } ]

a(n) = { my(m=n+1); while ((m^2 + 1)%(n^2 + 1) != 0, m++); m } \\ Harry J. Smith, Nov 03 2009

More terms from Robert G. Wilson v, Dec 11 2001

Further terms from Franklin T. Adams-Watters, Oct 17 2006

cp's OEIS Frontend

A117522 Numbers k such that L(2*k + 1) is prime, where L(m) is a Lucas number.

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A269253 Smallest prime in the sequence s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n + 1 (or -1 if no such prime exists).

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A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)*binomial(k-floor((j+1)/2), floor(j/2))*(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

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A299071 Union_{odd primes p, n >= 3} {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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A299100 Indices k such that s_k(4) is a (probable) prime, 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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A299101 Indices of (probable) primes in A030221.

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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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A066885 a(n) = (prime(n)^2 + 1)/2.

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A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.