A081264 -id:A081264 - cp's OEIS frontend

A327651 Composite numbers k coprime to 8 such that k divides Pell(k - Kronecker(8,k)), Pell = A000129.

35, 169, 385, 779, 899, 961, 1121, 1189, 2419, 2555, 2915, 3107, 3827, 6083, 6265, 6441, 6601, 6895, 6965, 7801, 8119, 8339, 9179, 9809, 9881, 10403, 10763, 10835, 10945, 13067, 14027, 14111, 15179, 15841, 18241, 18721, 19097, 20833, 20909, 22499, 23219, 24727, 26795, 27869, 27971

Offset: 1

Jianing Song, Sep 20 2019

nonn

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that a condition similar to (a) holds for k, where m = 2.
If k is not required to be coprime to m^2 + 4 (= 8), then there are 1232 such k <= 10^5 and 4973 such k <= 10^6, while there are only 83 terms <= 10^5 and 245 terms <= 10^6 in this sequence.
Also composite numbers k coprime to 8 such that A214028(k) divides k - Kronecker(8,k).

			Pell(36) = 21300003689580 is divisible by 35, so 35 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

             m                       m=1           m=2       m=3
k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  this seq  A327653
k | x(k)-Kronecker(m^2+4,k)        A049062       A099011   A327654
            both                   A212424       A327652   A327655
* k is composite and coprime to m^2 + 4.
Cf. A000129, A214028, A091337 ({Kronecker(8,n)}).

pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
isA327651(n)=!isprime(n) && !pellmod(n-kronecker(8,n), n) && gcd(n,8)==1 && n>1

A327652 Intersection of A099011 and A327651.

Original entry on oeis.org

169, 385, 961, 1121, 3827, 6265, 6441, 6601, 7801, 8119, 10945, 13067, 15841, 18241, 19097, 20833, 24727, 27971, 29953, 31417, 34561, 35459, 37345, 38081, 39059, 42127, 45961, 47321, 49105, 52633, 53041, 55969, 56953, 58241, 62481, 74305, 79361, 81361, 84587, 86033, 86241, 101311, 107801

Offset: 1

Jianing Song, Sep 20 2019

nonn

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that conditions similar to (a) and (b) hold for k simultaneously, where m = 2.

If k is not required to be coprime to m^2 + 4 (= 8), then there are 1190 such k <= 10^5 and 4847 such k <= 10^6, while there are only 41 terms <= 10^5 and 119 terms <= 10^6 in this sequence.

			169 divides Pell(168) as well as Pell(169) - 1, so 169 is a term.

Daniel Suteu, Table of n, a(n) for n = 1..10000

             m                       m=1           m=2       m=3
k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  A327651   A327653
k | x(k)-Kronecker(m^2+4,k)        A049062       A099011   A327654
            both                   A212424       this seq  A327655
* k is composite and coprime to m^2 + 4.
Cf. A000129, A091337 ({Kronecker(8,n)}).

pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
isA327652(n)=!isprime(n) && pellmod(n, n)==kronecker(8,n) && !pellmod(n-kronecker(8,n), n) && gcd(n,8)==1 && n>1

A327654 Composite numbers k coprime to 13 such that k divides A006190(k) - Kronecker(13,k).

Original entry on oeis.org

4, 8, 9, 119, 399, 649, 1023, 1179, 1189, 1199, 1881, 2703, 3519, 4081, 4187, 5151, 7055, 7361, 10349, 12871, 13833, 14041, 15519, 16109, 18639, 22593, 23479, 24769, 26937, 28421, 29007, 31631, 34111, 34997, 38503, 41441, 44671, 48577, 50545, 51711, 53823, 56279, 57407, 58081, 59081

Offset: 1

Jianing Song, Sep 20 2019

nonn

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that a condition similar to (b) holds for k, where m = 3.

If k is not required to be coprime to m^2 + 4 (= 13), then there are 352 such k <= 10^5, and 1457 such k <= 10^6, while there are only 54 terms <= 10^5 and 148 terms <= 10^6 in this sequence.

			A006190(8) = 3927 == Kronecker(13,8) (mod 8), so 8 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..2000

             m                      m=1            m=2      m=3
k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  A327651  A327653
k | x(k)-Kronecker(m^2+4,k)        A049062       A099011  this seq
            both                   A212424       A327652  A327655
* k is composite and coprime to m^2 + 4.
Cf. A006190, A011583 ({Kronecker(13,n)}).

seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327654(n)=!isprime(n) && seqmod(n, n)==kronecker(13,n) && gcd(n,13)==1 && n>1

A327655 Intersection of A327653 and A327654.

Original entry on oeis.org

119, 649, 1189, 4187, 12871, 14041, 16109, 23479, 24769, 28421, 31631, 34997, 38503, 41441, 48577, 50545, 56279, 58081, 59081, 61447, 75077, 91187, 95761, 96139, 116821, 127937, 146329, 148943, 150281, 157693, 170039, 180517, 188501, 207761, 208349, 244649, 281017, 311579, 316409

Offset: 1

Jianing Song, Sep 20 2019

nonn

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that conditions similar to (a) and (b) hold for k simultaneously, where m = 2.

If k is not required to be coprime to m^2 + 4 (= 13), then there are 322 such k <= 10^5 and 1381 such k <= 10^6, while there are only 24 terms <= 10^5 and 72 terms <= 10^6 in this sequence.

			119 divides A006190(120) as well as A006190(119) + 1, so 119 is a term.

             m                      m=1            m=2      m=3
k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  A327651  A327653
k | x(k)-Kronecker(m^2+4,k)        A049062       A099011  A327654
            both                   A212424       A327652  this seq
* k is composite and coprime to m^2 + 4.
Cf. A006190, A011583 ({Kronecker(13,n)}).

seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327655(n)=!isprime(n) && seqmod(n, n)==kronecker(13,n) && !seqmod(n-kronecker(13,n), n) && gcd(n,13)==1 && n>1

A340118 Odd composite integers m such that A000045(2*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

323, 377, 609, 1891, 3081, 3827, 4181, 5777, 5887, 6601, 6721, 8149, 10877, 11663, 13201, 13601, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 27323, 28441, 28623, 30889, 32509, 34561, 34943, 35207, 39203, 40501

Offset: 1

Ovidiu Bagdasar, Dec 28 2020

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.

The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=1, D=5 and k=2, while U(m) is A000045(m) (Fibonacci sequence).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).

Cf. A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Select[Range[3, 50000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 5], 1] - 1, #] &]

A340119 Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.

Original entry on oeis.org

9, 27, 63, 81, 99, 119, 153, 243, 567, 649, 729, 759, 891, 903, 1071, 1189, 1377, 1431, 1539, 1763, 1881, 1953, 2133, 2187, 3599, 3897, 4187, 4585, 5103, 5313, 5559, 5589, 5819, 6561, 6681, 6831, 6993, 8019, 8127, 8829, 8855, 9639, 9999, 10611, 11135, 11691, 11961

Offset: 1

Ovidiu Bagdasar, Dec 28 2020

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4. The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=3, D=13 and k=2, while U(m) is A006190(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A006190, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).

Cf. A340118 (a=1, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Select[Range[3, 12000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 13], 3] - 1, #] &]

A340235 Odd composite integers m such that A000045(3*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

9, 27, 161, 341, 901, 1107, 1281, 1853, 2241, 2529, 4181, 5473, 5611, 5777, 6119, 6721, 7587, 8307, 9729, 10877, 11041, 12209, 13201, 13277, 14981, 15251, 16771, 17567, 20591, 20769, 20801, 22827, 23323, 24921, 27403, 28421, 29489, 33001, 34561, 38529, 38801

Offset: 1

Ovidiu Bagdasar, Jan 01 2021

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=1, D=5 and k=3, while U(m) is A000045(m) (Fibonacci sequence).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A340118 (a=1, b=-1, k=2).

Cf. A340236 (a=3, b=-1, k=3), A340237 (a=5, b=-1, k=3), A340238 (a=7, b=-1, k=3).

Select[Range[3, 40000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 5], 1] - 1, #] &]

A177086 Semiprimes k that divide Fibonacci(k-1).

Original entry on oeis.org

1891, 4181, 8149, 13201, 15251, 17711, 40501, 51841, 64079, 64681, 67861, 68251, 78409, 88601, 88831, 90061, 96049, 97921, 115231, 118441, 145351, 146611, 153781, 191351, 197209, 218791, 219781, 254321, 272611, 302101, 303101

Offset: 1

Jonathan Vos Post, Dec 09 2010

This is the semiprime (A001358) analog of A045468. Now A045468 has a very simple characterization: it consists of the primes ending in 1 or 9. Can one say anything about the present sequence?

			46368/23 = 2016 = 2^5 * 3^2 * 7 so (24-1) | Fibonacci(24) but 24 is not semiprime, so is not in the sequence.
a(1) = 1891 = 31 * 61 is not in the sequence because 1891 divides Fibonacci(1891-1) = Fibonacci(1890).
a(21) = 146611 = 271 * 541 because 146611 | Fibonacci(146610).

Chai Wah Wu, Table of n, a(n) for n = 1..837

Cf. A000040, A000045, A001358, A069106, A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

Cf. A177745 (semiprimes k that divide Fibonacci(k+1)).

Select[Range[310000],PrimeOmega[#]==2 && Divisible[Fibonacci[#-1],#]&] (* Harvey P. Dale, May 02 2016 *)

{k: k is in A001358 and k|A000045(k-1)} = A069106 INTERSECTION A001358.

A177745 Semiprimes k that divide Fibonacci(k+1).

Original entry on oeis.org

323, 377, 3827, 5777, 10877, 11663, 18407, 19043, 23407, 25877, 27323, 34943, 39203, 51983, 53663, 60377, 75077, 86063, 94667, 100127, 113573, 121103, 121393, 161027, 162133, 182513, 195227, 200147, 231703, 240239, 250277, 294527, 306287, 345913, 381923, 429263, 430127, 454607, 500207, 507527, 548627, 569087, 600767, 635627, 636707, 685583, 697883, 736163, 753377, 775207, 828827, 851927, 948433, 983903

Offset: 1

Jonathan Vos Post, Dec 12 2010

Data from T. D. Noe.

			a(1) = 323 = 17 * 19 because it is semiprime (product of two prime A000040), and 323 divides F(324) = 23041483585524168262220906489642018075101617466780496790573690289968, with dividend 2^4 * 3^5 * 53 * 107 * 109 * 2269 * 3079 * 4373 * 5779 * 19441 * 11128427 * 62650261 * 1828620361 * 6782976947987.

Cf. A177086, A000045, A001358, A069106, A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

With[{semis=Select[Range[1000000],PrimeOmega[#]==2&]},Select[semis, Divisible[Fibonacci[#+1],#]&]] (* Harvey P. Dale, Aug 20 2012 *)

{k: k is in A001358 and k|A000045(k+1)}.

A214434 Composite numbers k such that k divides Fibonacci(k+1) or Fibonacci(k-1) and 2^(k-1) == 1 (mod k).

Original entry on oeis.org

6601, 13981, 30889, 68101, 219781, 252601, 332949, 399001, 512461, 642001, 721801, 722261, 741751, 852841, 873181, 1024651, 1141141, 1193221, 1207361, 1533601, 1690501, 1735841, 1857241, 1909001, 2085301, 2100901, 2165801, 2603381, 2704801, 2757241, 3186821, 3568661

Offset: 1

Gary Detlefs, Jul 17 2012

nonn

Pseudoprimes to a criterion for primality which tests that
1. k divides Fibonacci(k+1) or Fibonacci(k-1) (see A182554, A081264), and
2. 2^(k-1) == 1 (mod k) (see A001567).
All terms appear to be congruent to 1 or -1 (mod 5).
Terms that are not congruent to 1 or -1 (mod 5): 22711873, 40160737, 55462177, ... . - Amiram Eldar, Sep 12 2022

			6601 is in the sequence because the 6600th Fibonacci number is divisible by 6601 and 2^6600 = 1 mod 6601.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..671 from Giovanni Resta)

Cf. A182554, A081264, A001567.

with(combinat):f:= n-> fibonacci(n): for n from 1 to 2000000 do if(f(n+1) mod n = 0 or f(n-1) mod n = 0) and 2^(n-1) mod n = 1 and not isprime(n) then print(n) fi od;

Select[Range[1, 4*10^6, 2], CompositeQ[#] && PowerMod[2, # - 1, #] == 1 && (Divisible[Fibonacci[# - 1], #] || Divisible[Fibonacci[# + 1], #]) &] (* Amiram Eldar, Sep 12 2022 *)

Intersection of A182554 and A001567.

cp's OEIS Frontend

A327651 Composite numbers k coprime to 8 such that k divides Pell(k - Kronecker(8,k)), Pell = A000129.

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A327652 Intersection of A099011 and A327651.

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A327654 Composite numbers k coprime to 13 such that k divides A006190(k) - Kronecker(13,k).

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A327655 Intersection of A327653 and A327654.

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A340118 Odd composite integers m such that A000045(2*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

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A340119 Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.

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A340235 Odd composite integers m such that A000045(3*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

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A177086 Semiprimes k that divide Fibonacci(k-1).

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