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A023212
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Primes p such that 4*p+1 is also prime.
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22
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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
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson
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EXTENSIONS
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Name edited by Michel Marcus, Nov 27 2020
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STATUS
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approved
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