The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A023212 Primes p such that 4*p+1 is also prime. 22
 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 EXTENSIONS Name edited by Michel Marcus, Nov 27 2020 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 11 00:03 EDT 2021. Contains 342877 sequences. (Running on oeis4.)