A112860 -id:A112860 - 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, ", ")))

A092751 Primes of the form (2*n)!/(n!)^2 - 1.

Original entry on oeis.org

5, 19, 251, 48619, 155117519, 30067266499541039, 6637553085023755473070799, 399608854866744452032002440111, 5717214010165655645594487649236004008072121335004636113518216597999

Offset: 1

Jorge Coveiro, Apr 12 2004

nonn

Cf. A000984, A066726.

Cf. A075840 = n such that (2*n)!/(n!)^2-1 is prime, A112860 = primes of the form (2*n)!/(n!)^2+1.

Binomial[2#, # ] - 1 & /@ Select[ Range[150], PrimeQ[(2#)!/#!^2 - 1] &] (* Robert G. Wilson v, Apr 14 2004 *)

Corrected and extended by Robert G. Wilson v, Apr 14 2004

A116515 a(n) = the period of the Fibonacci numbers modulo p divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime.

Original entry on oeis.org

1, 2, 4, 2, 1, 4, 4, 1, 2, 1, 1, 4, 2, 2, 2, 4, 1, 4, 2, 1, 4, 1, 2, 4, 4, 1, 2, 2, 4, 4, 2, 1, 4, 1, 4, 1, 4, 2, 2, 4, 1, 1, 1, 4, 4, 1, 1, 2, 2, 1, 4, 1, 2, 1, 4, 2, 4, 1, 4, 2, 2, 4, 2, 1, 4, 4, 1, 4, 2, 1, 4, 1, 2, 4, 1, 2, 4, 4, 2, 2, 1, 4, 1, 4, 1, 2, 2, 4, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 2, 2, 1

Offset: 1

Nick Krempel, Mar 24 2006

Conditions on p_n mod 4 and mod 5 restrict possible values of a(n). The unknown (?) case is p = 1 mod 4 and (5|p) = 1, equivalently, p = 1 or 9 mod 20, where {1, 2, 4} all occur.

Number of zeros in fundamental period of Fibonacci numbers mod prime(n). [From T. D. Noe, Jan 14 2009]

			a(4) = 2, as 7 is the 4th prime, the Fibonacci numbers mod 7 have period 16, the first Fibonacci number divisible by 7 is F(8) = 21 = 3*7 and 16 / 8 = 2.
One period of the Fibonacci numbers mod 7 is 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, which has two zeros. Hence a(4)=2. [From _T. D. Noe_, Jan 14 2009]

Cf. A060305, A001602.

Cf. A112860, A053027, A053028 (primes producing 1, 2 and 4 zeros) [From T. D. Noe, Jan 14 2009]

a(n) = A060305(n) / A001602(n). a(n) is always one of {1, 2, 4}.

a(n) = A001176(prime(n)) [From T. D. Noe, Jan 14 2009]

cp's OEIS Frontend

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

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

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

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

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

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

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A092751 Primes of the form (2*n)!/(n!)^2 - 1.

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