A023212 Primes p such that 4*p+1 is also prime.

3, 7, 13, 37, 43, 67, 73, 79, 97, 127, 139, 163, 193, 199, 277, 307, 373, 409, 433, 487, 499, 577, 619, 673, 709, 727, 739, 853, 883, 919, 997, 1033, 1039, 1063, 1087, 1093, 1123, 1129, 1297, 1327, 1423, 1429, 1453, 1543, 1549, 1567, 1579, 1597, 1663, 1753

Offset: 1

Author

David W. Wilson

Keywords

• nonn

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 an 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.

Crossrefs

Cf. A001122, A005384, A043297, A088730.

Cf. A005098, A090866.

Cf. A182265, A182434.

Programs

Magma

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

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 *)

PARI

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

Formula

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

Extensions

Name edited by Michel Marcus, Nov 27 2020