

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
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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*(8k1).  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



