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A023212 Numbers n such that n and 4n + 1 are both prime. 21
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 (list; graph; refs; listen; history; text; internal format)
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

Solutions of the equation (4*n + 1)' + n' = 2, where n' is the arithmetic derivative of n. - Paolo P. Lava, Oct 31 2012

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

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

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 *)

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

Cf. A001122, A005384, A043297, A088730.

Cf. A005098, A090866.

Cf. A182265, A182434.

Sequence in context: A222187 A084611 A078454 * A106952 A106951 A106057

Adjacent sequences:  A023209 A023210 A023211 * A023213 A023214 A023215

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified October 20 20:24 EDT 2019. Contains 328273 sequences. (Running on oeis4.)