A024894 -id:A024894 - cp's OEIS frontend

A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

211, 281, 421, 461, 521, 691, 881, 991, 1031, 1151, 1511, 1601, 1871, 1951, 2221, 2591, 3001, 3251, 3571, 3851, 4021, 4391, 4441, 4481, 4621, 4651, 4691, 4751, 4871, 5081, 5281, 5381, 5531, 5591, 5641, 5801, 5881, 6011, 6101, 6211, 6271, 6491, 6841

Offset: 1

N. J. A. Sloane

From A.H.M. Smeets, Nov 15 2023: (Start)
Mean gap size between two consecutive terms at p: ~ 20*log(p) (see E. Lehmer).
In E. Lehmer, Artiads characterized, she counted in the table on p. 122 the primes p for which p == 1 (mod 5) instead of all primes. As a result, in the corollary on p. 121, the 20% becomes 5% (or 1/20 instead of 1/5). (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..24903 (first 1000 terms from T. D. Noe)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138 (annotated scanned copy).
Michael J. Mossinghoff and Christopher Pinner, Prime power order circulant determinants, arXiv:2205.12439 [math.NT], 2022. See Type 2 primes on p. 3.
H. W. Lloyd Tanner, On the Binomial Equation x^p-1=0: Quinquisection, Proc. London Math. Soc., 18 (1886-1887), 214-234.
H. W. Lloyd Tanner, On Complex Primes formed with the Fifth Roots of Unity, Proc. London Math. Soc., 24 (1892-1893), 223-262.

Cf. A047650, A000045, A024894, subsequence of A030430.

See also A270798 (a subsequence), A270800.

a001583 n = a001583_list !! (n-1)
a001583_list = filter
   (\p -> mod (a000045 $ div (p - 1) 5) p == 0) a030430_list
-- Reinhard Zumkeller, Aug 15 2013

Select[ Prime[ Range[1000]], Mod[#, 5] == 1 && Divisible[ Fibonacci[(# - 1)/5], #] &] (* Jean-François Alcover, Jun 22 2012 *)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
list(lim)=my(v=List()); forprime(p=11,lim, if(p%5==1 && fibmod(p\5,p)==0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 06 2017

From A.H.M. Smeets, Nov 15 2023: (Start)
Equals {prime(m): A296240(m) == 0 (mod 5)}.
a(n) ~ prime(20*n). (End)

More terms from James Sellers, Jan 25 2000

Edited by N. J. A. Sloane, Apr 01 2016

A153329 Numbers k such that 5*k + 1 is not prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 37, 39, 40, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 87

Offset: 1

Vincenzo Librandi, Dec 23 2008

Numbers k such that (5*k)!/(5*k + 1) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the even terms in the following triangular array:
   *;
   *,  *;
   4,  *,  *;
   *,  *,  *, 16;
   *,  *,  *,  *, 24;
   *,  *, 18,  *,  *,  *;
   *,  *,  *,  *,  *,  *,  *;
  10,  *,  *,  *,  *, 44,  *,  *;
   *,  *,  *, 34,  *,  *,  *,  *, 72;
   *,  *,  *,  *, 46,  *,  *,  *,  *, 88;
   *,  *, 32,  *,  *,  *,  *, 78,  *,  *,  *;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A024894, A014076, A153170, A045751, A095277, A153088, A153343.

[n: n in [0..150] | not IsPrime(5*n + 1)]; // Vincenzo Librandi, Jan 12 2013

for n from 0 to 100 do
if irem(factorial(5*n), 5*n+1) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[0, 200], !PrimeQ[5*# + 1]&] (* Vincenzo Librandi, Jan 12 2013 *)

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010

0 added by Arkadiusz Wesolowski, Aug 03 2011

A111223 Numbers n such that 5*n + 2 is prime.

Original entry on oeis.org

0, 1, 3, 7, 9, 13, 19, 21, 25, 27, 31, 33, 39, 45, 51, 55, 61, 63, 67, 69, 73, 79, 91, 93, 97, 109, 111, 115, 117, 121, 123, 129, 135, 145, 151, 157, 159, 165, 171, 175, 177, 181, 187, 189, 193, 195, 199, 217, 219, 223, 237, 243, 247, 255, 259, 261, 265, 273, 285

Offset: 1

Parthasarathy Nambi, Oct 26 2005

			97 is in the sequence because 5*97 + 2 = 487 is prime.

T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley, New York, 2001, p. 410 (Theorem 34.8).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A024894.

[n: n in [0..350] | IsPrime(5*n+2)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[0, 1000], PrimeQ[5 # + 2] &] (* Vincenzo Librandi, May 20 2014 *)
Table[If[PrimeQ[5p+2], Mod[5^(-1) Fibonacci[5p], 5p+2],  Unevaluated[Sequence[]]], {p, 0, 250}] (* Rigoberto Florez, Mar 02 2020 *)
Select[(#-2)/5&/@Prime[Range[250]],IntegerQ] (* Harvey P. Dale, Sep 27 2023 *)

is(n)=isprime(5*n+2) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = F(p-2)/5 mod p, where p is the n-th prime number such that p==2 (mod 5) and F(m) is m-th Fibonacci number. - Rigoberto Florez, Mar 02 2020

A111225 Numbers n such that 5*n + 8 is prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 31, 33, 37, 43, 45, 51, 55, 57, 61, 69, 73, 75, 85, 87, 91, 99, 103, 111, 117, 121, 127, 129, 133, 135, 145, 147, 153, 163, 169, 171, 175, 189, 195, 201, 205, 211, 217, 219, 223, 229, 231, 237, 241, 243, 255, 259, 273, 283, 285, 289

Offset: 1

Parthasarathy Nambi, Oct 26 2005

nonn

			If n=103 then 5*n + 8 = 523 (prime).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..100000] |IsPrime(5*n+8)]; // Vincenzo Librandi, Sep 03 2010

Select[Range[300],PrimeQ[5#+8]&] (* Harvey P. Dale, Oct 31 2011 *)

is(n)=isprime(5*n+8) \\ Charles R Greathouse IV, Feb 17 2017

A111224 Numbers n such that 5*n + 7 is prime.

Original entry on oeis.org

0, 2, 6, 8, 12, 18, 20, 24, 26, 30, 32, 38, 44, 50, 54, 60, 62, 66, 68, 72, 78, 90, 92, 96, 108, 110, 114, 116, 120, 122, 128, 134, 144, 150, 156, 158, 164, 170, 174, 176, 180, 186, 188, 192, 194, 198, 216, 218, 222, 236, 242, 246, 254, 258, 260, 264, 272, 284

Offset: 1

Parthasarathy Nambi, Oct 26 2005

			If n=108 then 5*n + 7 = 547 (prime).

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..300] | IsPrime(5*n+7)]; // Vincenzo Librandi, Sep 03 2010

Select[Range[0,300],PrimeQ[5#+7]&] (* Harvey P. Dale, Jun 24 2018 *)

is(n)=isprime(5*n+7) \\ Charles R Greathouse IV, Jun 12 2017

A367970 Least k such that 5nk+1 is a prime.

Original entry on oeis.org

2, 1, 2, 2, 4, 1, 2, 1, 4, 2, 6, 1, 2, 1, 2, 3, 12, 2, 2, 1, 2, 3, 4, 2, 2, 1, 2, 2, 10, 1, 2, 4, 2, 6, 4, 1, 8, 1, 6, 2, 4, 1, 2, 3, 8, 2, 4, 1, 2, 1, 4, 2, 4, 1, 12, 1, 2, 5, 4, 2, 6, 1, 2, 2, 4, 1, 6, 3, 2, 2, 6, 5, 18, 4, 2, 2, 6, 3, 6, 1, 2, 2, 28, 1, 6

Offset: 1

Robert Price, Dec 17 2023

nonn

Robert Price, Table of n, a(n) for n = 1..5000

Cf. A016014, A024894.

A070849 lists the corresponding primes.

A367970 = {};
Do[k=1; While[!PrimeQ[5 n k+1], k++]; AppendTo[A367970 ,k], {n,85}];
A367970

a(n) = my(k=1); while (!isprime(5*n*k+1), k++); k; \\ Michel Marcus, Dec 17 2023

A111226 Numbers n such that 5*n + 12 is prime.

Original entry on oeis.org

1, 5, 7, 11, 17, 19, 23, 25, 29, 31, 37, 43, 49, 53, 59, 61, 65, 67, 71, 77, 89, 91, 95, 107, 109, 113, 115, 119, 121, 127, 133, 143, 149, 155, 157, 163, 169, 173, 175, 179, 185, 187, 191, 193, 197, 215, 217, 221, 235, 241, 245, 253, 257, 259, 263, 271, 283, 287

Offset: 1

Parthasarathy Nambi, Oct 26 2005

			If n=109 then 5*n + 12 = 557 (prime).

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..100000] |IsPrime(5*n+12)]; // Vincenzo Librandi, Nov 13 2010

Mathematica
```
Select[Range[300], PrimeQ[5 # + 12] &]
```

is(n)=isprime(5*n+12) \\ Charles R Greathouse IV, Jun 13 2017

A111230 Numbers k such that 5*k + 14 is prime.

Original entry on oeis.org

1, 3, 9, 13, 15, 19, 25, 27, 33, 37, 43, 45, 51, 67, 69, 73, 75, 79, 81, 85, 87, 93, 97, 99, 111, 117, 121, 129, 139, 141, 145, 151, 159, 163, 165, 169, 181, 183, 199, 201, 205, 207, 211, 219, 223, 243, 247, 249, 253, 255, 261, 277, 279, 283, 285, 289, 295, 297

Offset: 1

Parthasarathy Nambi, Oct 27 2005

			If k=111 then 5*k + 14 = 569 (prime).

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..100000] |IsPrime(5*n+14)]; // Vincenzo Librandi, Nov 13 2010

is(n)=isprime(5*n+14) \\ Charles R Greathouse IV, Jun 13 2017

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A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

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A153329 Numbers k such that 5*k + 1 is not prime.

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A111223 Numbers n such that 5*n + 2 is prime.

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A111225 Numbers n such that 5*n + 8 is prime.

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A111224 Numbers n such that 5*n + 7 is prime.

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Magma

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A367970 Least k such that 5*n*k+1 is a prime.

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Mathematica

PARI

A111226 Numbers n such that 5*n + 12 is prime.

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Magma

Mathematica

A367970 Least k such that 5nk+1 is a prime.