OFFSET

1,1

COMMENTS

If p > 3 is a Sophie Germain prime (A005384), p cannot be in this sequence, because all Germain primes greater than 3 are of the form 6k - 1, and then 4p + 1 = 3*(8k-1). - Enrique Pérez Herrero, Aug 15 2011

a(n), except 3, is of the form 6k+1. - Enrique Pérez Herrero, Aug 16 2011

According to Beiler: the integer 2 is a primitive root of all primes of the form 4p + 1 with p prime. - Martin Renner, Nov 06 2011

Chebyshev showed that 2 is a primitive root of all primes of the form 4p + 1 with p prime. - Jonathan Sondow, Feb 04 2013

Also solutions to the equation: floor(4/A000005(4*n^2+n)) = 1. - Enrique Pérez Herrero, Jan 12 2013

Prime numbers p such that p^p - 1 is divisible by 4*p + 1. - Gary Detlefs, May 22 2013

It appears that whenever (p^p - 1)/(4*p + 1) is integer, then this integer is even (see previous comment). - Alexander R. Povolotsky, May 23 2013

4p + 1 does not divide p^n + 1 for any n. - Robin Garcia, Jun 20 2013

Primes in this sequence of the form 4k+1 are listed in A113601. - Gary Detlefs, May 07 2019

There are no numbers with last digit 1 in this list (i.e., members of A030430) because primes p == 1 (mod 10) lead to 5|(4p+1) such that 4p+1 is not prime. - R. J. Mathar, Aug 13 2019

REFERENCES

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 102, nr. 5.

P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.

LINKS

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Grigory Ryabov, On schurity of the dihedral group D_(2p), arXiv:2308.14209 [math.GR], 2023.

Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.

Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.

FORMULA

Sum_{n>=1} 1/a(n) is in the interval (0.892962433, 1.1616905) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021

MAPLE

isA023212 := proc(n)

isprime(n) and isprime(4*n+1) ;

end proc:

for n from 1 to 1800 do

if isA023212(n) then

printf("%d, ", n) ;

end if;

end do: # R. J. Mathar, May 26 2013

MATHEMATICA

Select[Range[2000], PrimeQ[#] && PrimeQ[4# + 1] &] (* Alonso del Arte, Aug 15 2011 *)

Join[{3}, Select[Range[7, 2000, 6], PrimeQ[#] && PrimeQ[4# + 1] &]] (* Zak Seidov, Jan 21 2012 *)

Select[Prime[Range[300]], PrimeQ[4#+1]&] (* Harvey P. Dale, Oct 17 2021 *)

PROG

(Magma) [n: n in [0..1000] | IsPrime(n) and IsPrime(4*n+1)] // Vincenzo Librandi, Nov 20 2010

(PARI) forprime(p=2, 1800, if(Mod(p, 4*p+1)^p==1, print1(p", \n"))) \\ Alexander R. Povolotsky, May 23 2013

CROSSREFS

KEYWORD

nonn

AUTHOR

EXTENSIONS

Name edited by Michel Marcus, Nov 27 2020

STATUS

approved