login
Primes congruent to {1, 4} mod 5.
46

%I #78 May 22 2022 09:48:44

%S 11,19,29,31,41,59,61,71,79,89,101,109,131,139,149,151,179,181,191,

%T 199,211,229,239,241,251,269,271,281,311,331,349,359,379,389,401,409,

%U 419,421,431,439,449,461,479,491

%N Primes congruent to {1, 4} mod 5.

%C Rational primes that decompose in the field Q(sqrt(5)). - _N. J. A. Sloane_, Dec 26 2017

%C These are also primes p that divide Fibonacci(p-1). - _Jud McCranie_

%C Primes ending in 1 or 9. - _Lekraj Beedassy_, Oct 27 2003

%C Also primes p such that p divides 5^(p-1)/2 - 4^(p-1)/2. - _Cino Hilliard_, Sep 06 2004

%C Primes p such that the polynomial x^2-x-1 mod p has 2 distinct zeros. - _T. D. Noe_, May 02 2005

%C Same as A038872, apart from the term 5. - _R. J. Mathar_, Oct 18 2008

%C Appears to be the primes p such that p^6 mod 210 = 1. - _Gary Detlefs_, Dec 29 2011

%C Primes in A047209, also in A090771. - _Reinhard Zumkeller_, Jan 07 2012

%C 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

%C 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

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

%H Reinhard Zumkeller, <a href="/A045468/b045468.txt">Table of n, a(n) for n = 1..10000</a>

%H Caleb Ji, Tanya Khovanova, Robin Park, and Angela Song, <a href="http://arxiv.org/abs/1509.06093">Chocolate Numbers</a>, arXiv:1509.06093 [math.CO], 2015.

%H Caleb Ji, Tanya Khovanova, Robin Park, and Angela Song, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Khovanova/khova9.html">Chocolate Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.1.7.

%H <a href="https://oeis.org/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>

%p 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

%t 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 *)

%t Select[Prime[Range[200]],MemberQ[{1,4},Mod[#,5]]&] (* _Vincenzo Librandi_, Aug 13 2012 *)

%o (PARI) list(lim)=select(n->n%5==1||n%5==4,primes(primepi(lim))) \\ _Charles R Greathouse IV_, Jul 25 2011

%o (Haskell)

%o a045468 n = a045468_list !! (n-1)

%o a045468_list = [x | x <- a047209_list, a010051 x == 1]

%o -- _Reinhard Zumkeller_, Jan 07 2012

%o (Magma) [ p: p in PrimesUpTo(1000) | p mod 5 in {1,4} ]; // _Vincenzo Librandi_, Aug 13 2012

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

%Y Subsequence of A123976.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_