A045468 Primes congruent to {1, 4} mod 5.

11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Rational primes that decompose in the field Q(sqrt(5)). - N. J. A. Sloane, Dec 26 2017

These are also primes p that divide Fibonacci(p-1). - Jud McCranie

Primes ending in 1 or 9. - Lekraj Beedassy, Oct 27 2003

Also primes p such that p divides 5^(p-1)/2 - 4^(p-1)/2. - Cino Hilliard, Sep 06 2004

Primes p such that the polynomial x^2-x-1 mod p has 2 distinct zeros. - T. D. Noe, May 02 2005

Same as A038872, apart from the term 5. - R. J. Mathar, Oct 18 2008

Appears to be the primes p such that p^6 mod 210 = 1. - Gary Detlefs, Dec 29 2011

Primes in A047209, also in A090771. - Reinhard Zumkeller, Jan 07 2012

Primes p such that p does not divide Sum_{i=1..p} Fibonacci(i)^2. The sum is A001654(p). - Arkadiusz Wesolowski, Jul 23 2012

Primes congruent to {1, 9} mod 10. Legendre symbol (5, a(n)) = +1. For prime 5 this symbol (5, 5) is set to 0, and (5, prime) = -1 for prime == {3, 7} (mod 10), given in A003631. - Wolfdieter Lang, Mar 05 2021

References

• Hardy and Wright, An Introduction to the Theory of Numbers, Chap. X, p. 150, Oxford University Press, Fifth edition.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Caleb Ji, Tanya Khovanova, Robin Park, and Angela Song, Chocolate Numbers, arXiv:1509.06093 [math.CO], 2015.
• Caleb Ji, Tanya Khovanova, Robin Park, and Angela Song, Chocolate Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.1.7.
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

Cf. A030430, A030433, A064739, A038872, A010051, A003631.

Subsequence of A123976.

Programs

Haskell

a045468 n = a045468_list !! (n-1)
a045468_list = [x | x <- a047209_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

Magma

[ p: p in PrimesUpTo(1000) | p mod 5 in {1,4} ]; // Vincenzo Librandi, Aug 13 2012

Maple

for n from 1 to 500 do if(isprime(n)) and (n^6 mod 210=1) then print(n) fi od;  # Gary Detlefs, Dec 29 2011

Mathematica

lst={};Do[p=Prime[n];If[Mod[p,5]==1||Mod[p,5]==4,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
Select[Prime[Range[200]],MemberQ[{1,4},Mod[#,5]]&] (* Vincenzo Librandi, Aug 13 2012 *)

PARI

list(lim)=select(n->n%5==1||n%5==4,primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011