# On Generalized Lucas Pseudoprimality of Level k

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**generalized Lucas**sequence ${\{{U}_{n}(a,b)\}}_{n\ge 0}$ and its companion, the

**generalized Pell–Lucas**sequence ${\{{V}_{n}(a,b)\}}_{n\ge 0}$, denoted by ${U}_{n}$ and ${V}_{n}$ for simplicity, are defined by

**Theorem**

**1**

**.**Let p be an odd prime, k a non-negative integer, and r an arbitrary integer. If $b=\pm 1$ and a is an integer such that $D={a}^{2}-4b>0$ is not a perfect square, then the sequences ${U}_{n}$ and ${V}_{n}$ defined by (1) and (2) satisfy the following relations

**Theorem**

**2**

**.**Let p be an odd prime, and let $k>0$ and a be integers so that $D={a}^{2}+4>0$ is not a perfect square. If ${U}_{n}={U}_{n}(a,-1)$ and ${V}_{n}={V}_{n}(a,-1)$, then we have

- (1)
- ${U}_{kp-\left(\frac{D}{p}\right)}\equiv {U}_{k-1}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p);$
- (2)
- ${V}_{kp-\left(\frac{D}{p}\right)}\equiv \left(\frac{D}{p}\right){V}_{k-1}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p).$

**Theorem**

**3**

**.**Let p be an odd prime, and let $k>0$ and a be integers so that $D={a}^{2}-4>0$ is not a perfect square. If ${U}_{n}={U}_{n}(a,1)$ and ${V}_{n}={V}_{n}(a,1)$, then we have

- (1)
- ${U}_{kp-\left(\frac{D}{p}\right)}\equiv \left(\frac{D}{p}\right){U}_{k-1}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p);$
- (2)
- ${V}_{kp-\left(\frac{D}{p}\right)}\equiv {V}_{k-1}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p).$

**Definition**

**1.**

**generalized Lucas pseudoprime of parameters**a

**and**b if $gcd(n,b)=1$ and n divides ${U}_{n-\left(\frac{D}{n}\right)}$, where $\left(\frac{D}{n}\right)$ is the Jacobi symbol.

**Definition**

**2.**

**weak generalized Lucas pseudoprime**of parameters a and b.

**Definition**

**3.**

**generalized Bruckman–Lucas pseudoprime of parameters**a

**and**b if $n\mid {V}_{n}(a,b)-a$.

**Proposition**

**1.**

- (1)
- ${U}_{n}\equiv \left(\frac{D}{n}\right)\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n);$
- (2)
- ${V}_{n}\equiv {V}_{1}=a\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n)$;
- (3)
- ${U}_{n-\left(\frac{D}{n}\right)}\equiv {U}_{0}=0\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n);$
- (4)
- ${V}_{n-\left(\frac{D}{n}\right)}\equiv 2{b}^{\frac{1-\left(\frac{D}{n}\right)}{2}}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n)$ (valid whenever $gcd(n,D)=1$).

## 2. Fibonacci and Lucas Pseudoprimes of Level k

#### 2.1. Fibonacci Pseudoprimes of Level k

**Fibonacci pseudoprime**if $n\mid {F}_{n-\left(\frac{n}{5}\right)}$. The even Fibonacci pseudoprimes are indexed as A141137 in the OEIS [15], while the odd Fibonacci pseudoprimes indexed as A081264 start with the terms

**Fibonacci pseudoprime of level**k if it satisfies

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

**Example**

**1.**

^{®}. We have

**Remark**

**2.**

#### 2.2. Lucas Pseudoprimes of Level k

**Bruckman–Lucas pseudoprime**. The sequence is indexed in the OEIS [15] as A005845, and begins with

**Lucas pseudoprimes of level**k as the composite integers n satisfying the relation

**Proposition**

**3.**

**Proof.**

**Example**

**2.**

## 3. Generalized Lucas Pseudoprimes of Level k

#### 3.1. Jacobi’s Symbol

#### 3.2. Results for $b=-1$

- (1)
- ${U}_{kp-\left(\frac{p}{D}\right)}\equiv {U}_{k-1}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p);$
- (2)
- ${V}_{kp-\left(\frac{p}{D}\right)}\equiv \left(\frac{p}{D}\right){V}_{k-1}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p)$.

**Definition**

**4.**

**generalized Lucas pseudoprime of level ${k}^{-}$ and parameter**a if

**generalized Pell–Lucas pseudoprime of level ${k}^{-}$ and parameter**a if

**Theorem**

**4.**

- (1)
- Reference [19] (Theorem 4.3). If $n\in {\mathcal{U}}_{1}^{-}(a)$, then $n\in {\mathcal{U}}_{2}^{-}(a)$ if and only if $n\mid {U}_{n}^{2}-1$.
- (2)
- Reference [19] (Theorem 4.6). If $n\in {\mathcal{V}}_{1}^{-}(a)$ and $gcd(a,n)=1$, then $n\in {\mathcal{V}}_{2}^{-}(a)$ if and only if $n\mid {U}_{n}^{2}-1$.

**Remark**

**3.**

**Remark**

**4.**

- $a=1$: None found for $n\le 50000$ (see also, Remark 1);
- $a=3$: $9,63,99,153,1071,1881,1953,9999,13833,16191$;
- $a=5$: None found for $n\le 15000$;
- $a=7$: $49,147,245,637,833,1127,1225,2499,3185,3479,4753,5537,15925$.

- $a=1$: $323,377,1891,3827,6601,8149,11663,13981,17119,17711,18407,19043$;
- $a=3$: $1763,3599,5559,6681,12095,12403,12685,14279,15051,19043$;
- $a=5$: $15,45,91,135,143,1547,1573,1935,2015,6543,8099,10403,10905$;
- $a=7$: $35,65,175,391,455,575,1247,1295,1763,1775,2275,2407,3367,4199,4579$.

**Remark**

**5.**

- $a=1$: $323,377,1891,3827,6601,8149,11663,13981,17119,17711,18407,19043$;
- $a=3$: $1763,3599,5559,6681,12095,12403,12685,14279,15051,19043$;
- $a=5$: $15,45,91,135,143,1547,1573,1935,2015,6543,8099,10403,10905$;
- $a=7$: $35,65,175,391,455,575,1247,1295,1763,1775,2275,2407,3367,4199,4579$.

- $a=1$: None found for $n\le 50000$;
- $a=3$: None found for $n\le 20000$;
- $a=5$: $18901,19601,19951$;
- $a=7$: None found for $n\le 17000$.

**Conjecture**

**1.**

**Example**

**3.**

- The set ${\mathcal{U}}_{2}^{-}(1)$ gives A340118 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 323,377,609,1891,3081,3827,4181,5777,5887,6601,6721,8149,10877,11663,13201,\hfill \\ & 13601,13981,15251,17119,17711,18407,19043,23407,25877,27323,28441,28623,\hfill \\ & 30889,32509,34561,34943,35207,39203,40501,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{3}^{-}(1)$ is A340235 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,27,161,341,901,1107,1281,1853,2241,2529,4181,5473,5611,5777,6119,6721,\hfill \\ & 7587,8307,9729,10877,11041,12209,13201,13277,14981,15251,16771,17567,\dots .\hfill \end{array}$$
- The sequence ${\mathcal{V}}_{3}^{-}(1)$ is given by A339724 and starts with the elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,21,161,341,901,1281,1853,3201,4181,5473,5611,5777,6119,6721,9729,10877,\hfill \\ & 11041,12209,12441,13201,14981,15251,16771,17941,20591,20769,20801,\dots .\hfill \end{array}$$

**Example**

**4.**

- The set ${\mathcal{U}}_{1}^{-}(3)$ recovers pseudoprimes indexed as A327653 in [15], starting with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 119,649,1189,1763,3599,4187,5559,6681,12095,12403,12685,12871,12970,14041,\hfill \\ & 14279,15051,16109,19043,22847,23479,24769,26795,28421,30743,30889,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{2}^{-}(3)$ gives A340119 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,27,63,81,99,119,153,243,567,649,729,759,891,903,1071,1189,1377,1431,1539,\hfill \\ & 1763,1881,1953,2133,2187,3599,3897,4187,4585,5103,5313,5559,5589,5819,\hfill \\ & 6561,6681,6831,6993,8019,8127,8829,8855,9639,9999,10611,11135,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{3}^{-}(3)$ is indexed as A340236 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,119,121,187,327,345,649,705,1003,1089,1121,1189,1881,2091,2299,3553,4187,\hfill \\ & 5461,5565,5841,6165,6485,7107,7139,7145,7467,7991,8321,8449,11041,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{1}^{-}(3)$ recovers A339126, and starts with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,25,49,119,121,289,361,529,649,833,841,961,1089,1189,1369,1681,1849,1881,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& 2023,2209,2299,2809,3025,3481,3721,4187,4489,5041,5329,6241,6889,7139,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{2}^{-}(3)$ giving A339518, has the first elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 15,75,105,119,165,255,375,649,1189,1635,1763,1785,1875,2233,2625,3599,3815,\hfill \\ & 4125,4187,5475,5559,5887,6375,6601,6681,7905,8175,9265,9375,9471,11175,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{3}^{-}(3)$ is given by A339725 and starts with the elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,27,119,133,145,165,205,261,341,393,649,693,705,901,945,1121,1173,1189,\hfill \\ & 1353,1431,1485,1881,2133,2805,3201,3605,3745,4187,5173,5461,5841,5945,\dots .\hfill \end{array}$$

**Example**

**5.**

- The set ${\mathcal{U}}_{1}^{-}(5)$ recovers the entry A340095 in [15], starting with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,15,27,45,91,121,135,143,1547,1573,1935,2015,6543,6721,8099,10403,10877,\hfill \\ & 10905,13319,13741,13747,14399,14705,16109,16471,18901,19043,19109,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{2}^{-}(5)$ gives A340120 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,15,25,27,45,75,91,121,125,135,143,147,175,225,275,325,375,441,483,625,\hfill \\ & 675,735,755,1125,1323,1547,1573,1875,1935,2015,2205,2275,2485,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{3}^{-}(5)$ is indexed as A340237 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,27,33,35,65,81,99,121,221,243,297,363,513,585,627,705,729,891,1089,1539,\hfill \\ & 1541,1881,2145,2187,2299,2673,3267,3605,4181,4573,4579,5265,5633,6721,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{1}^{-}(5)$ recovers A339127, and starts with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,25,27,49,81,121,169,175,225,243,289,325,361,529,637,729,961,1225,1331,\hfill \\ & 1369,1539,1681,1849,2025,2209,2809,3025,3481,3721,4225,4489,5041,5329,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{2}^{-}(5)$ giving A339519, has the first elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,15,27,39,45,91,117,121,135,143,195,287,351,507,585,741,1521,1547,1573,\hfill \\ & 1755,1935,2015,2067,2535,2601,3157,3227,3445,3505,3519,3731,4563,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{3}^{-}(5)$ is given by A339726 and starts with the elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,25,27,33,35,45,65,81,99,117,121,161,175,221,225,297,325,363,585,645,705,\hfill \\ & 825,891,1089,1281,1539,1541,1881,2025,2133,2145,2181,2299,2325,2925,\dots .\hfill \end{array}$$

**Example**

**6.**

- The set ${\mathcal{U}}_{1}^{-}(7)$ recovers the entry A340096 in [15], starting with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 25,35,51,65,91,175,325,391,455,575,1247,1295,1633,1763,1775,1921,2275,\hfill \\ & 2407,2599,2651,3367,4199,4579,4623,5629,6441,9959,10465,10825,10877,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{2}^{-}(7)$ gives A340121 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 25,35,39,49,51,65,91,147,175,245,301,325,343,391,455,507,575,605,637,663,\hfill \\ & 741,833,897,903,935,1127,1205,1225,1247,1295,1505,1595,1633,1715,1763,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{3}^{-}(7)$ is indexed as A340238 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,25,27,51,91,105,153,185,225,289,325,425,459,481,513,747,867,897,925,\hfill \\ & 945,1001,1189,1299,1469,1633,1785,1921,2241,2245,2599,2601,2651,2769,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{1}^{-}(7)$ recovers A339128, and starts with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,25,49,51,91,121,125,153,169,289,325,361,441,529,625,637,833,841,867,961,\hfill \\ & 1183,1225,1369,1633,1681,1849,1921,2209,2599,2601,2651,3481,3721,4225,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{2}^{-}(7)$ giving A339520, has the first elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 25,35,51,65,75,91,105,175,203,325,391,455,575,645,861,1247,1275,1295,\hfill \\ & 1633,1763,1775,1785,1875,1921,2275,2407,2415,2599,2625,2651,3045,3367,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{3}^{-}(7)$ is given by A339727 and starts with the elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,25,49,51,69,91,105,143,145,153,185,221,225,325,339,391,425,441,481,\hfill \\ & 637,645,705,805,833,897,925,1001,1173,1189,1207,1225,1281,1299,1365,\dots .\hfill \end{array}$$

**Conjecture**

**2.**

#### 3.3. Results for $b=1$

- (1)
- ${U}_{kp-\left(\frac{p}{D}\right)}\equiv \left(\frac{p}{D}\right){U}_{k-1}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p);$
- (2)
- ${V}_{kp-\left(\frac{p}{D}\right)}\equiv {V}_{k-1}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p).$

**Definition**

**5.**

**generalized Lucas pseudoprime of level ${k}^{+}$ and parameter**a if

**generalized Pell–Lucas pseudoprime of level ${k}^{+}$ and parameter**a if

**Theorem**

**5.**

- (1)
- Reference [19] (Theorem 4.9). If $n\in {\mathcal{U}}_{1}^{+}(a)$, then $n\in {\mathcal{U}}_{2}^{+}(a)$ if and only if $n\mid {U}_{n}^{2}-1$.
- (2)
- Reference [19] (Theorem 4.12). If $n\in {\mathcal{V}}_{1}^{+}(a)$ and $gcd(a,n)=1$, then $n\in {\mathcal{V}}_{2}^{+}(a)$ if and only if $n\mid {U}_{n}^{2}-1$.

**Remark**

**6.**

**Remark**

**7.**

**Remark**

**8.**

- $a=3$: $9,63,423,2871,2961,8001$;
- $a=5$: $25,275,425,575,775,6325,6575,9775,13175,17825$;
- $a=7$: $49,1127,2303$

- $a=3$: $21,329,451,861,1081,1819,2033,2211,3653,4089,5671,8557,11309$,$13861,14701,17513,17941,19951,20473$;
- $a=5$: $115,253,391,713,715,779,935,1705,2627,2893,2929,3281,4141,5191$,$5671,7739,8695,11815,12121,17963$;
- $a=7$: $1771,7471,7931,15449$.

**Example**

**7.**

^{®}. We have

**Remark**

**9.**

- $a=3$: $21,329,451,861,1081,1819,2033,2211,3653,4089,5671,8557,11309$,$13861,14701,17513,17941,19951,20473$;
- $a=5$: $115,253,391,713,715,779,935,1705,2627,2893,2929,3281,4141,5191$,$5671,7739,11815,12121,17963$;
- $a=7$: $1771,7471,7931,15449$.

**Conjecture**

**3.**

**Example**

**8.**

- The set ${\mathcal{U}}_{1}^{+}(3)$ recovers the entry A340097 in [15], starting with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 21,323,329,377,451,861,1081,1819,1891,2033,2211,3653,3827,4089,4181,5671,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& 5777,6601,6721,8149,8557,10877,11309,11663,13201,13861,13981,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{2}^{+}(3)$ recovers A340122 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,21,27,63,81,189,243,323,329,351,377,423,451,567,729,783,861,891,963,1081,\hfill \\ & 1701,1743,1819,1891,1967,2033,2187,2211,2871,2889,2961,3321,3653,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{3}^{+}(3)$ is indexed as A340239 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,49,63,141,161,207,323,341,377,441,671,901,1007,1127,1281,1449,1853,\hfill \\ & 1891,2071,2303,2407,2501,2743,2961,3827,4181,4623,5473,5611,5777,6119,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{1}^{+}(3)$ recovers A339129, and starts with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,49,63,121,169,289,323,361,377,441,529,841,961,1127,1369,1681,1849,1891,\hfill \\ & 2209,2303,2809,2961,3481,3721,3751,3827,4181,4489,4901,4961,5041,5329,5491,\hfill \\ & 5777,6137,6241,6601,6721,6889,7381,7921,8149,9409,10201,10609,10877,10933,\hfill \\ & 11449,11663,11881,12769,13201,13981,14027,15251,16129,17119,17161,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{2}^{+}(3)$ giving A339521, has the first elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 21,203,323,329,377,451,609,861,1001,1081,1183,1547,1729,1819,1891,2033,\hfill \\ & 2211,2821,3081,3549,3653,3827,4089,4181,4669,5671,5777,5887,6601,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{3}^{+}(3)$ is given by A339728 and starts with the elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 9,21,27,63,161,189,207,261,287,323,341,377,671,783,861,901,987,1007,\hfill \\ & 1107,1269,1281,1287,1449,1853,1891,2071,2241,2407,2431,2501,2529,2567,\hfill \\ & 2743,2961,3201,3827,4181,4623,5029,5473,5611,5777,5781,6119,6601,\dots .\hfill \end{array}$$

**Conjecture**

**4.**

**Conjecture**

**5.**

**Example**

**9.**

- The set ${\mathcal{U}}_{1}^{+}(5)$ recovers the entry A340098 in [15], starting with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 115,253,391,527,551,713,715,779,935,1705,1807,1919,2627,2893,2929,3281,\hfill \\ & 4033,4141,5191,5671,5777,5983,6049,6479,7645,7739,8695,9361,11663,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{2}^{+}(5)$ recovers A340123 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 25,115,125,253,275,391,425,505,527,551,575,625,713,715,775,779,935,1705,\hfill \\ & 1807,1919,2525,2627,2875,2893,2929,3125,3281,4033,4141,5191,5555,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{3}^{+}(5)$ is indexed as A340240 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 55,407,527,529,551,559,965,1199,1265,1633,1807,1919,1961,3401,3959,4033,\hfill \\ & 4381,5461,5777,5977,5983,6049,6233,6439,6479,7141,7195,7645,7999,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{1}^{+}(5)$ recovers A339130, and starts with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 25,121,169,275,289,361,527,529,551,575,841,961,1369,1681,1807,1849,1919,\hfill \\ & 2209,2783,2809,3025,3481,3721,4033,4489,5041,5329,5777,5983,6049,6241,\hfill \\ & 6479,6575,6889,7267,7645,7921,8959,8993,9361,9409,9775,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{2}^{+}(5)$ giving A339522, has the first elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 95,115,145,253,391,527,551,713,715,779,935,1045,1615,1705,1805,1807,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& 1919,2185,2627,2755,2893,2929,2945,3281,4033,4141,4205,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{3}^{+}(5)$ is given by A339729 and starts with the elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 25,55,85,115,155,187,253,275,341,407,527,551,559,575,851,925,1199,1265,\hfill \\ & 1633,1775,1807,1919,1961,2123,2507,2635,2641,2725,\dots .\hfill \end{array}$$

**Example**

**10.**

- The set ${\mathcal{U}}_{1}^{+}(7)$ recovers the entry A340099 in [15], starting with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 323,329,377,451,1081,1771,1819,1891,2033,3653,3827,4181,5671,5777,6601,\hfill \\ & 6721,7471,7931,8149,8557,10877,11309,11663,13201,13861,13981,14701,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{2}^{+}(7)$ recovers A340124 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 49,323,329,343,377,451,1081,1127,1771,1819,1891,2033,2303,2401,3653,3827,\hfill \\ & 4181,5671,5777,6601,6721,7471,7931,8149,8557,9691,10877,11309,\dots .\hfill \end{array}$$
- The set ${\mathcal{U}}_{3}^{+}(7)$ is indexed as A340241 and its first elements are$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 161,323,329,341,377,451,671,901,1007,1079,1081,1271,1819,1853,1891,2033,\hfill \\ & 2071,2209,2407,2461,2501,2743,3653,3827,4181,4843,5473,5611,5671,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{1}^{+}(7)$ recovers A339131, and starts with$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 49,121,169,289,323,329,361,377,451,529,841,961,1081,1127,1369,1681,1819,\hfill \\ & 1849,1891,2033,2209,2303,2809,3481,3653,3721,3751,3827,4181,4489,4901,\hfill \\ & 4961,5041,5329,5491,5671,5777,6137,6241,6601,6721,6889,7381,7921,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{2}^{+}(7)$ giving A339523, has the first elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 91,203,323,329,377,451,1001,1081,1183,1547,1729,1771,1819,1891,1967,2033,\hfill \\ & 2093,2639,2821,3197,3311,3653,3731,3827,4181,4669,\dots .\hfill \end{array}$$
- The set ${\mathcal{V}}_{3}^{+}(7)$ is given by A339730 and starts with the elements$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& 49,161,287,323,329,341,377,451,671,737,901,1007,1079,1081,1127,1271,1363,\hfill \\ & 1541,1819,1853,1891,1927,2033,2071,2303,2407,2431,2461,2501,2567,2743,\dots .\hfill \end{array}$$

**Conjecture**

**6.**

## 4. Conclusions and Further Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Andrica, D.; Bagdasar, O.
On Generalized Lucas Pseudoprimality of Level *k*. *Mathematics* **2021**, *9*, 838.
https://doi.org/10.3390/math9080838

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Andrica D, Bagdasar O.
On Generalized Lucas Pseudoprimality of Level *k*. *Mathematics*. 2021; 9(8):838.
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Andrica, Dorin, and Ovidiu Bagdasar.
2021. "On Generalized Lucas Pseudoprimality of Level *k*" *Mathematics* 9, no. 8: 838.
https://doi.org/10.3390/math9080838