A261580 -id:A261580 - 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, ", ")))

A309586 Primes p with 1 zero in a fundamental period of A006190 mod p.

Original entry on oeis.org

2, 3, 23, 43, 53, 61, 79, 101, 103, 107, 127, 131, 139, 173, 179, 191, 199, 211, 251, 263, 277, 283, 311, 347, 367, 419, 433, 439, 443, 467, 491, 503, 523, 547, 563, 569, 571, 599, 607, 647, 659, 677, 719, 727, 751, 757, 823, 829, 859, 881, 883, 887, 907

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A322906(p) = 1.
For p > 2, p is in this sequence if and only if A175182(p) == 2 (mod 4), and if and only if A322907(p) == 2 (mod 4). For a proof of the equivalence between A322906(p) = 1 and A322907(p) == 2 (mod 4), see Section 2 of my link below.
This sequence contains all primes congruent to 3, 23, 27, 35, 43, 51 modulo 52. This corresponds to case (3) for k = 11 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

Jianing Song, Table of n, a(n) for n = 1..1200
Jianing Song, Lucas sequences and entry point modulo p

Cf. A175182, A322907.
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 | this seq
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  | A309585 | A309593
* and also A053032 U {2}

forprime(p=2, 900, if(A322906(p)==1, print1(p, ", ")))

A309587 Primes p with 2 zeros in a fundamental period of A006190 mod p.

Original entry on oeis.org

7, 11, 17, 19, 31, 47, 59, 67, 71, 83, 113, 151, 163, 167, 223, 227, 239, 257, 271, 307, 313, 331, 337, 359, 379, 383, 431, 463, 479, 487, 499, 521, 587, 601, 619, 631, 641, 643, 673, 683, 691, 739, 743, 787, 809, 811, 827, 839, 863, 947, 967, 983

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A322906(p) = 2.
For p > 2, p is in this sequence if and only if 8 divides A175182(p), and if and only if 4 divides A322907(p). For a proof of the equivalence between A322906(p) = 2 and 4 dividing A322907(p), see Section 2 of my link below.
This sequence contains all primes congruent to 7, 11, 15, 19, 31, 47 modulo 52. This corresponds to case (2) for k = 11 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

Jianing Song, Table of n, a(n) for n = 1..1200
Jianing Song, Lucas sequences and entry point modulo p

Cf. A006190, A175182, A322907.
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 | this seq
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  | A309585 | A309593
* and also A053032 U {2}

forprime(p=2, 1000, if(A322906(p)==2, print1(p, ", ")))

A309588 Primes p with 4 zeros in a fundamental period of A006190 mod p.

Original entry on oeis.org

5, 13, 29, 37, 41, 73, 89, 97, 109, 137, 149, 157, 181, 193, 197, 229, 233, 241, 269, 281, 293, 317, 349, 353, 373, 389, 397, 401, 409, 421, 449, 457, 461, 509, 541, 557, 577, 593, 613, 617, 653, 661, 701, 709, 733, 761, 769, 773, 797, 821, 853, 857, 877

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A322906(p) = 4.
For p > 2, p is in this sequence if and only if A175182(p) == 4 (mod 8), and if and only if A322907(p) is odd. For a proof of the equivalence between A322906(p) = 4 and A322907(p) being odd, see Section 2 of my link below.
This sequence contains all primes congruent to 5, 21, 33, 37, 41, 45 modulo 52. This corresponds to case (1) for k = 11 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

Jianing Song, Table of n, a(n) for n = 1..1200
Jianing Song, Lucas sequences and entry point modulo p

Cf. A006190, A175182, A322907.
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 | this seq
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  | A309585 | A309593
* and also A053032 U {2}

forprime(p=2, 900, if(A322906(p)==4, print1(p, ", ")))

A309591 Numbers k with 1 zero in a fundamental period of A006190 mod k.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 23, 27, 36, 43, 46, 53, 54, 61, 69, 79, 81, 86, 92, 101, 103, 106, 107, 108, 122, 127, 129, 131, 138, 139, 158, 159, 162, 172, 173, 179, 183, 191, 199, 202, 206, 207, 211, 212, 214, 237, 243, 244, 251, 254, 258, 262, 263, 276

Offset: 1

Jianing Song, Aug 10 2019

nonn

Numbers k such that A322906(k) = 1.

The odd numbers k satisfy A175182(k) == 2 (mod 4).

Jianing Song, Table of n, a(n) for n = 1..3000

Cf. A175182.
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 | this seq
Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
* and also A053032 U {2}

for(k=1, 300, if(A322906(k)==1, print1(k, ", ")))

A309592 Numbers k with 2 zeros in a fundamental period of A006190 mod k.

Original entry on oeis.org

7, 8, 11, 14, 15, 16, 17, 19, 20, 21, 22, 24, 28, 30, 31, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 47, 48, 49, 51, 52, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 75, 76, 77, 78, 80, 83, 84, 85, 87, 88, 90, 91, 93, 94, 95, 96, 98, 99, 100, 102, 104

Offset: 1

Jianing Song, Aug 10 2019

nonn

Numbers k such that A322906(k) = 2.
This sequence contains all numbers k such that 4 divides A322907(k). As a consequence, this sequence contains all numbers congruent to 7, 11, 15, 19, 31, 47 modulo 52.
This sequence contains all odd numbers k such that 8 divides A175182(k).

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

Cf. A175182, A322907.
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 | this seq
Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
* and also A053032 U {2}

for(k=1, 100, if(A322906(k)==2, print1(k, ", ")))

A309593 Numbers k with 4 zeros in a fundamental period of A006190 mod k.

Original entry on oeis.org

5, 10, 13, 25, 26, 29, 37, 41, 50, 58, 65, 73, 74, 82, 89, 97, 109, 125, 130, 137, 145, 146, 149, 157, 169, 178, 181, 185, 193, 194, 197, 205, 218, 229, 233, 241, 250, 269, 274, 281, 290, 293, 298, 314, 317, 325, 338, 349, 353, 362, 365, 370, 373, 377

Offset: 1

Jianing Song, Aug 10 2019

nonn

Numbers k such that A322906(k) = 4.

Also numbers k such that A214027(k) is odd.

Jianing Song, Table of n, a(n) for n = 1..2000

Cf. A322907.
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  | A309585 | this seq
* and also A053032 U {2}

for(k=1, 400, if(A322906(k)==4, print1(k, ", ")))

A299959 Least prime factor of (4^(2n+1)+1)/5, a(0) = 1.

Original entry on oeis.org

1, 13, 5, 29, 13, 397, 53, 5, 137, 229, 13, 277, 5, 13, 107367629, 5581, 13, 5, 149, 13, 10169, 173, 5, 3761, 29, 13, 15358129, 5, 13, 1181, 733, 13, 5, 269, 13, 569, 293, 5, 29, 317, 13, 997, 5, 13, 1069, 29, 13, 5, 389, 13, 809, 41201, 5, 857, 5669, 13, 58309, 5, 13, 29, 397, 13, 5, 509, 13

Offset: 0

M. F. Hasler, Feb 22 2018

nonn

The range of this sequence with a(0) = 1 omitted, {5, 13, 29, 53, 137, ...}, appears to be a subset of A261580 (and of the Pythagorean primes A002144). Is there a smaller superset sequence in OEIS? - M. F. Hasler, Jan 07 2025

			For n = 0, A299960(0) = (4^1+1)/5 = 5/5 = 1, therefore we let a(0) = 1.
For n = 1, A299960(1) = (4^3+1)/5 = 65/5 = 13 is prime, therefore a(1) = 13.
For n = 2, A299960(2) = (4^5+1)/5 = 1025/5 = 205 = 5*41, therefore a(2) = 5.

Chai Wah Wu, Table of n, a(n) for n = 0..715

Cf. A299960, A052539.

a(n)=A020639(4^(2*n+1)\5+1) \\ Using factor(...)[1,1] requires complete factorization and is much less efficient for large n.

a(n) = A020639(A299960(n)) = A020639(A052539(2n+1)/5).
a(n) = 5 iff n = 2 (mod 5); otherwise, a(n) = 13 if n = 1 (mod 3).
Otherwise, a(n) = 29 if n = 3 (mod 7), else a(n) = 53 if n = 6 (mod 13), else a(n) = 137 if n = 8 (mod 17), else a(n) = 149 if n = 18 (mod 27), else a(n) = 173 if n = 21 (mod 43), etc... - M. F. Hasler, Jan 07 2025

A261581 Primes such that z(p) is not divisible by 4 where z(n) is A214028(n), the smallest k such that n divides A000129(k), the k-th Pell number.

Original entry on oeis.org

2, 5, 7, 13, 23, 29, 31, 37, 41, 47, 53, 61, 71, 79, 101, 103, 109, 127, 137, 149, 151, 157, 167, 173, 181, 191, 197, 199, 223, 229, 239, 263, 269, 271, 277, 293, 311, 313, 317, 349, 353, 359, 367, 373, 383, 389, 397, 409, 421, 431, 439, 457, 461, 463, 479, 487

Offset: 1

Michel Marcus, Aug 25 2015

nonn

			The smallest Pell number divisible by the prime 2 has index 2, which is not divisible by 4, so 2 is in the sequence.

Bernadette Faye and Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015.

Cf. A000129, A214028, A261580.

pell(n) = polcoeff(Vec(x/(1-2*x-x^2) + O(x^(n+1))), n);
z(n) = {k=1; while (pell(k) % n, k++); k;}
lista(nn) = {forprime(p=2, nn, if (z(p) % 4, print1(p, ", ")););}

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A309586 Primes p with 1 zero in a fundamental period of A006190 mod p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A309587 Primes p with 2 zeros in a fundamental period of A006190 mod p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A309588 Primes p with 4 zeros in a fundamental period of A006190 mod p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A309591 Numbers k with 1 zero in a fundamental period of A006190 mod k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A309592 Numbers k with 2 zeros in a fundamental period of A006190 mod k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A309593 Numbers k with 4 zeros in a fundamental period of A006190 mod k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A299959 Least prime factor of (4^(2n+1)+1)/5, a(0) = 1.

Views

Author

Keywords

Comments

Examples

Links