A214028 -id:A214028 - cp's OEIS frontend

A309584 Numbers k with 2 zeros in a fundamental period of A000129 mod k.

3, 6, 9, 10, 11, 12, 15, 17, 18, 19, 21, 22, 26, 27, 30, 33, 34, 35, 36, 38, 39, 42, 43, 44, 45, 50, 51, 54, 55, 57, 58, 59, 60, 63, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 81, 83, 84, 85, 86, 87, 89, 90, 91, 93, 95, 97, 99, 102, 105, 106, 107, 108, 110

Offset: 1

Jianing Song, Aug 10 2019

nonn

Numbers k such that A214027(k) = 2.
This sequence contains all numbers k such that 4 divides A214028(k). As a consequence, this sequence contains all numbers congruent to 3 modulo 8.
This sequence contains all odd numbers k such that 8 divides A175181(k).

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A175181, A214028.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2    |   m=3
-----------------------------+----------+----------+---------
The sequence {x(n)}          | A000045  | A000129  | A006190
The sequence {w(k)}          | A001176  | A214027  | A322906
Primes p such that w(p) = 1  | A112860* | A309580  | A309586
Primes p such that w(p) = 2  | A053027  | A309581  | A309587
Primes p such that w(p) = 4  | A053028  | A261580  | A309588
Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
Numbers k such that w(k) = 2 | A053030  | this seq | A309592
Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
* and also A053032 U {2}

for(k=1, 100, if(A214027(k)==2, print1(k, ", ")))

A309585 Numbers k with 4 zeros in a fundamental period of A000129 mod k.

Original entry on oeis.org

5, 13, 25, 29, 37, 53, 61, 65, 101, 109, 125, 137, 145, 149, 157, 169, 173, 181, 185, 197, 229, 265, 269, 277, 293, 305, 317, 325, 349, 373, 377, 389, 397, 421, 461, 481, 505, 509, 521, 541, 545, 557, 569, 593, 613, 625, 653, 661, 677, 685, 689, 701, 709

Offset: 1

Jianing Song, Aug 10 2019

nonn

Numbers k such that A214027(k) = 4.

Also numbers k such that A214028(k) is odd.

Jianing Song, Table of n, a(n) for n = 1..1000

Cf. A214028.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2    |   m=3
-----------------------------+----------+----------+---------
The sequence {x(n)}          | A000045  | A000129  | A006190
The sequence {w(k)}          | A001176  | A214027  | A322906
Primes p such that w(p) = 1  | A112860* | A309580  | A309586
Primes p such that w(p) = 2  | A053027  | A309581  | A309587
Primes p such that w(p) = 4  | A053028  | A261580  | A309588
Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
Numbers k such that w(k) = 4 | A053029  | this seq | A309593
* and also A053032 U {2}

for(k=1, 700, if(A214027(k)==4, print1(k, ", ")))

A322907 Entry points for the 3-Fibonacci numbers A006190.

Original entry on oeis.org

1, 3, 2, 6, 3, 6, 8, 6, 6, 3, 4, 6, 13, 24, 6, 12, 8, 6, 20, 6, 8, 12, 22, 6, 15, 39, 18, 24, 7, 6, 32, 24, 4, 24, 24, 6, 19, 60, 26, 6, 7, 24, 42, 12, 6, 66, 48, 12, 56, 15, 8, 78, 26, 18, 12, 24, 20, 21, 12, 6, 30, 96, 24, 48, 39, 12, 68, 24, 22, 24, 72, 6

Offset: 1

Jianing Song, Jan 05 2019

nonn

a(n) is the smallest k > 0 such that n divides A006190(k).
a(n) is also called the rank of A006190(n) modulo n.
For primes p == 1, 9, 17, 25, 29, 49 (mod 52), a(p) divides (p - 1)/2.
For primes p == 3, 23, 27, 35, 43, 51 (mod 52), a(p) divides p - 1, but a(p) does not divide (p - 1)/2.
For primes p == 5, 21, 33, 37, 41, 45 (mod 52), a(p) divides (p + 1)/2.
For primes p == 7, 11, 15, 19, 31, 47 (mod 52), a(p) divides p + 1, but a(p) does not divide (p + 1)/2.
a(n) <= (12/7)*n for all n, where the equality holds if and only if n = 2*7^e, e >= 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Jianing Song)

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = k*x(n+1) + x(n). Then the periods, ranks and the ratios of the periods to the ranks modulo a given integer n are given by:
k = 1: A001175 (periods), A001177 (ranks), A001176 (ratios).
k = 2: A175181 (periods), A214028 (ranks), A214027 (ratios).
k = 3: A175182 (periods), this sequence (ranks), A322906 (ratios).
Cf. A006190.

A006190(m) = ([3, 1; 1, 0]^m)[2, 1]
a(n) = my(i=1); while(A006190(i)%n!=0, i++); i

a(m*n) = a(m)*a(n) if gcd(m, n) = 1.
For odd primes p, a(p^e) = p^(e-1)*a(p) if p^2 does not divide a(p). Any counterexample would be a 3-Wall-Sun-Sun prime.
a(2^e) = 3 if e = 1, 6 if e = 2 and 3*2^(e-2) if e >= 3. a(13^e) = 13^e, e >= 1.

A246556 a(n) = smallest prime which divides Pell(n) = A000129(n) but does not divide any Pell(k) for k

Original entry on oeis.org

2, 5, 3, 29, 7, 13, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 137, 199, 37, 19, 45697, 23, 229, 1153, 1549, 79, 53, 113, 44560482149, 31, 61, 665857, 52734529, 103, 1800193921, 73, 593, 9369319, 389, 241, 1746860020068409, 4663, 11437, 43, 6481, 47, 3761, 97, 293, 45245801, 101, 22307, 68480406462161287469, 7761799, 109, 1535466241

Offset: 2

Eric Chen, Nov 15 2014

nonn

First differs from A264137 (Largest prime factor of the n-th Pell number) at n=17; see Example section. - Jon E. Schoenfield, Dec 10 2016

			a(2) = 2 because Pell(2) = 2 and Pell(k) < 2 for k < 2.
a(4) = 3 because Pell(4) = 12 = 2^2 * 3, but 2 is not a primitive prime factor since Pell(2) = 2, so therefore 3 is the primitive prime factor.
a(5) = 29 because Pell(5) = 29, which is prime.
a(6) = 7 because Pell(6) = 70 = 2 * 5 * 7, but neither 2 nor 5 is a primitive prime factor, so therefore 7 is the primitive prime factor.
a(17) = 137 because Pell(17) = 1136689 = 137 * 8297, and both of them are primitive factors, we choose the smallest. (Pell(17) is the smallest Pell number with more than one primitive prime factor.)

Table of n, a(n) for n = 2..630
R. D. Carmichael, On the numerical factors of the arithmetic forms α^n ± β^n, Annals of Math., 15 (1/4) (1913), 30-70.

Cf. A001578 (for Fibonacci(n)), A000129 (Pell numbers), A008555, A086383, A096650, A120947, A175181, A214028, A264137.

prms={}; Table[f=First/@FactorInteger[Pell[n]]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, First[p]], {n, 36}]

a(n) >= 2 for all n >= 2, by Carmichael's theorem. - Jonathan Sondow, Dec 08 2017

Edited by N. J. A. Sloane, Nov 29 2014
Terms up to a(612) in b-file added by Sean A. Irvine, Sep 23 2019
Terms a(613)-a(630) in b-file added by Max Alekseyev, Aug 26 2021

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

Original entry on oeis.org

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

A271782 Smallest n-Wall-Sun-Sun prime.

Original entry on oeis.org

13, 241, 2, 3, 191, 5, 2, 3, 2683

Offset: 2

Felix Fröhlich, Apr 18 2016

A prime p is a k-Wall-Sun-Sun prime iff p^2 divides F_k(pi_k(p)), where F_k(n) is the k-Fibonacci numbers, i.e., a Lucas sequence of first kind with (P,Q) = (k,-1), and pi_k(p) is the Pisano period of k-Fibonacci numbers modulo p (cf. A001175, A175181-A175185).
For prime p > 2 not dividing k^2 + 4, it is a k-Wall-Sun-Sun prime iff p^2 | F_k(p-((k^2+4)/p)), where ((k^2+4)/p) is the Kronecker symbol.
a(1) would be the smallest Wall-Sun-Sun prime whose existence is an open question.
a(12)..a(16) = 2, 3, 3, 29, 2. a(18)..a(33) = 3, 11, 2, 23, 3, 3, 2, 5, 79, 3, 2, 7, 23, 3, 2, 239. a(36)..a(38) = 2, 7, 17. a(40), a(41) = 2, 3. a(43)..a(46) = 5, 2, 3, 41. - R. J. Mathar, Apr 22 2016
a(17) = 1192625911, a(35) = 153794959, a(39) = 30132289567, a(47)..a(50) = 139703, 2, 3, 3. If they exist, a(11), a(34), a(42) are greater than 10^12. - Max Alekseyev, Apr 26 2016
Column k = 1 of table T(n, k) of k-th n-Wall-Sun-Sun prime (that table is not yet in the OEIS as a sequence). - Felix Fröhlich, Apr 25 2016
From Richard N. Smith, Jul 16 2019: (Start)
a(n) = 2 if and only if n is divisible by 4.
a(n) = 3 if and only if n == 5, 9, 13, 14, 18, 22, 23, 27, 31 (mod 36). (End)

Sergio Falcon and Ángel Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41:1 (2009), 497-504.
Richard N. Smith, Table of n, a(n) for n = 1..1024 (or 0 if unknown, searched up to 2^22)
Richard N. Smith, List of odd n-Wall-Sun-Sun primes <= 2^20 for 1<=n<=1024
Wikipedia, Wall-Sun-Sun prime

Cf. A001177, A039951, A113649, A113650, A113651, A214028, A237517, A237835, A241014, A244801, A253247, A268478.

A271782(k) = forprime(p=2,10^8, if( (([0,1;1,k]*Mod(1,p^2))^(p-kronecker(k^2+4,p)))[1,2]==0, return(p);); ); \\ Max Alekseyev, Apr 22 2016, corrected by Richard N. Smith, Jul 16 2019 to include p=2 and p divides k^2+4

a(4n) = 2.

Edited by Max Alekseyev, Apr 25 2016

A308949 a(n) is the greatest divisor of A000129(n) that is coprime to A000129(m) for all positive integers m < n.

Original entry on oeis.org

1, 2, 5, 3, 29, 7, 169, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 1136689, 199, 6625109, 1121, 45697, 8119, 225058681, 1153, 45232349, 47321, 7761797, 38081, 44560482149, 961, 259717522849, 665857, 52734529, 1607521, 1800193921, 13067, 51422757785981

Offset: 1

Jianing Song, Jul 02 2019

nonn

a(n) is squarefree unless n is of the form A214028(A238736(k)) = {7, 30, 1546462, ...}. The terms in A238736 are called 2-Wall-Sun-Sun primes.

			A000129(30) = 107578520350 = 2 * 5^2 * 7 * 29 * 31^2 * 41 * 269. We have 2, 7 divides A000129(6) = 70; 29, 41 divides A000129(10) = 2378; 5, 269 divides A000129(15) = 195025, but A000129(m) is coprime to 31 for all 1 <= m < 30, so a(30) = 31^2 = 961.

Cf. A000129, A008555, A178763, A214028, A238736.

nmax = 40;
pell = {1, 2};
pp = {1, 2};
Do[s = 2*pell[[-1]] + pell[[-2]];
  AppendTo[pell, s];
  AppendTo[pp, s/Times @@ pp[[Most[Divisors[n]]]]], {n, 3, nmax}];
a[2] = 2;
a[n_] := pp[[n]]/GCD[pp[[n]], n];
Array[a, nmax] (* Jean-François Alcover, Jul 06 2019, after T. D. Noe in A008555 *)

T(n) = ([2, 1; 1, 0]^n)[2, 1]
b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
a(n) = if(n==2, 2, b(n)/gcd(n, b(n)))

a(n) = A008555(n) / gcd(A008555(n), n) if n != 2.

A173257 Pell sequence entry points for primes == 1 (mod 4).

Original entry on oeis.org

3, 7, 8, 5, 19, 10, 27, 31, 36, 44, 48, 51, 55, 28, 17, 75, 79, 87, 91, 96, 9, 23, 116, 40, 64, 15, 139, 140, 49, 78, 159, 28, 175, 22, 187, 39, 199, 200, 102, 211, 216, 224, 114, 231, 255, 65, 271, 279, 71, 16

Offset: 1

R. J. Mathar, Jul 07 2012

R. J. Mathar, Table of n, a(n) for n = 1..1000
Neville Robbins, On Pell numbers of the form p*X^2, where p is prime, Fib Quart. 22 (4) (1984) 340, Table 2.

A173257 := proc(n)
        A214028(A002144(n)) ;
end proc:
seq(A173257(n),n=1..50) ;

a(n) = A214028(A002144(n)).

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A309584 Numbers k with 2 zeros in a fundamental period of A000129 mod k.

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A309585 Numbers k with 4 zeros in a fundamental period of A000129 mod k.

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A322907 Entry points for the 3-Fibonacci numbers A006190.

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A246556 a(n) = smallest prime which divides Pell(n) = A000129(n) but does not divide any Pell(k) for k

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

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A271782 Smallest n-Wall-Sun-Sun prime.

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A308949 a(n) is the greatest divisor of A000129(n) that is coprime to A000129(m) for all positive integers m < n.

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A173257 Pell sequence entry points for primes == 1 (mod 4).

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