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%I #128 Feb 04 2024 01:10:30
%S 3,7,13,37,43,67,73,79,97,127,139,163,193,199,277,307,373,409,433,487,
%T 499,577,619,673,709,727,739,853,883,919,997,1033,1039,1063,1087,1093,
%U 1123,1129,1297,1327,1423,1429,1453,1543,1549,1567,1579,1597,1663,1753
%N Primes p such that 4*p+1 is also prime.
%C 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
%C a(n), except 3, is of the form 6k+1. - _Enrique Pérez Herrero_, Aug 16 2011
%C 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
%C Chebyshev showed that 2 is a primitive root of all primes of the form 4p + 1 with p prime. - _Jonathan Sondow_, Feb 04 2013
%C Also solutions to the equation: floor(4/A000005(4*n^2+n)) = 1. - _Enrique Pérez Herrero_, Jan 12 2013
%C Prime numbers p such that p^p - 1 is divisible by 4*p + 1. - _Gary Detlefs_, May 22 2013
%C 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
%C 4p + 1 does not divide p^n + 1 for any n. - _Robin Garcia_, Jun 20 2013
%C Primes in this sequence of the form 4k+1 are listed in A113601. - _Gary Detlefs_, May 07 2019
%C 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
%D Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 102, nr. 5.
%D P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.
%H Enrique Pérez Herrero, <a href="/A023212/b023212.txt">Table of n, a(n) for n = 1..5000</a>
%H Grigory Ryabov, <a href="https://arxiv.org/abs/2308.14209">On schurity of the dihedral group D_(2p)</a>, arXiv:2308.14209 [math.GR], 2023.
%H Rosemary Sullivan and Neil Watling, <a href="https://www.emis.de/journals/INTEGERS/papers/n65/n65.Abstract.html">Independent divisibility pairs on the set of integers from 1 to n</a>, INTEGERS, Vol. 13 (2013), Article A65.
%H Samuel S. Wagstaff, Jr., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Wagstaff/wag4.html">Sum of Reciprocals of Germain Primes</a>, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.
%F Sum_{n>=1} 1/a(n) is in the interval (0.892962433, 1.1616905) (Wagstaff, 2021). - _Amiram Eldar_, Nov 04 2021
%p isA023212 := proc(n)
%p isprime(n) and isprime(4*n+1) ;
%p end proc:
%p for n from 1 to 1800 do
%p if isA023212(n) then
%p printf("%d,",n) ;
%p end if;
%p end do: # _R. J. Mathar_, May 26 2013
%t Select[Range[2000], PrimeQ[#] && PrimeQ[4# + 1] &] (* _Alonso del Arte_, Aug 15 2011 *)
%t Join[{3}, Select[Range[7, 2000, 6], PrimeQ[#] && PrimeQ[4# + 1] &]] (* _Zak Seidov_, Jan 21 2012 *)
%t Select[Prime[Range[300]],PrimeQ[4#+1]&] (* _Harvey P. Dale_, Oct 17 2021 *)
%o (Magma) [n: n in [0..1000] | IsPrime(n) and IsPrime(4*n+1)] // _Vincenzo Librandi_, Nov 20 2010
%o (PARI) forprime(p=2,1800,if(Mod(p,4*p+1)^p==1, print1(p", \n"))) // _Alexander R. Povolotsky_, May 23 2013
%Y Cf. A001122, A005384, A043297, A088730.
%Y Cf. A005098, A090866.
%Y Cf. A182265, A182434.
%K nonn
%O 1,1
%A _David W. Wilson_
%E Name edited by _Michel Marcus_, Nov 27 2020
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