A182265 -id:A182265 - cp's OEIS frontend

A023212 Primes p such that 4*p+1 is also prime.

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

Offset: 1

David W. Wilson

nonn

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
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 an 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

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.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Grigory Ryabov, On schurity of the dihedral group D_(2p), arXiv:2308.14209 [math.GR], 2023.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.

Cf. A001122, A005384, A043297, A088730.
Cf. A005098, A090866.
Cf. A182265, A182434.

[n: n in [0..1000] | IsPrime(n) and IsPrime(4*n+1)]; // Vincenzo Librandi, Nov 20 2010

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

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 *)
Select[Prime[Range[300]],PrimeQ[4#+1]&] (* Harvey P. Dale, Oct 17 2021 *)

forprime(p=2,1800,if(Mod(p,4*p+1)^p==1, print1(p", \n"))) \\ Alexander R. Povolotsky, May 23 2013

Sum_{n>=1} 1/a(n) is in the interval (0.892962433, 1.1616905) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021

Name edited by Michel Marcus, Nov 27 2020

A092816 Number of Sophie Germain primes less than 10^n.

Original entry on oeis.org

3, 10, 37, 190, 1171, 7746, 56032, 423140, 3308859, 26569515, 218116524, 1822848478, 15462601989, 132822315652

Offset: 1

Eric W. Weisstein, Mar 06 2004

Hardy-Littlewood conjecture: Number of Sophie Germain primes less than n ~ 2*C2*n/(log(n))^2, where C2 = 0.6601618158... is the twin prime constant (see A005597). The truth of the above conjecture would imply that there are an infinite number of Sophie Germain primes (which is also conjectured). - Robert G. Wilson v, Jan 31 2013

			The Sophie Germain primes up to 10 are 2 (since 5 is prime), 3 (since 7 is prime), and 5 (since 11 is prime), so a(1) = 3.

P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991, p. 228.

C. K. Caldwell, An amazing prime heuristic, Table 6.
Eric Weisstein's World of Mathematics, Sophie Germain Prime

Cf. A005384, A156874, A182265.

For 1 < n < 15, a(n) ~ e * (pi(2*10^n) - pi(10^n)) / (5*n - 5) where e is Napier's constant, see A001113 (we use n > 1 to avoid division by zero; whether the formula holds for any n > 14 is unknown). - Sergey Pavlov, Apr 07 2021 [This formula fails under the Hardy-Littlewood conjecture; the leading constant is wrong. - Charles R Greathouse IV, Aug 03 2023]

For any n, a(n) = qcc(x) - (10^n - pi(10^n) - pi(2 * 10^n + 1) + 1) where qcc(x) is the number of "common composite numbers" c <= 10^n such that both c and c' = 2*c + 1 are composite (trivial). - Sergey Pavlov, Apr 08 2021

a(10) computed by Eric W. Weisstein, Nov 02 2005
a(11)-a(12) from Donovan Johnson, Jun 19 2010
a(13)-a(14) from Giovanni Resta, Sep 04 2017

A182434 Number of primes p < n such that 4*p+1 is also prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Enrique Pérez Herrero, Apr 28 2012

nonn

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Cf. A023212, A182265.

Cf. A005384, A156660, A156874, A092816.

Accumulate[Table[Boole[PrimeQ[n]&&PrimeQ[4n+1]],{n,1,200}]]
Accumulate[If[AllTrue[{#,4#+1},PrimeQ],1,0]&/@Range[90]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 26 2015 *)

a(10^n) = A182265(n).

a(n) = sum(i=2..n, floor(phi(4*i^2+i)/(4*i^2-4*i))). - Enrique Pérez Herrero, May 02 2012.

A210684 Number of primes p < 10^n such that both 2p+1 and 4p+1 are composite.

Original entry on oeis.org

0, 7, 101, 864, 7365, 63331, 554839, 4931118, 44339730, 402709395, 3687732409, 34007530868

Offset: 1

Enrique Pérez Herrero, May 09 2012

nonn

Cf. A006880, A092816, A182265, A156660, A005384, A156874.

f[n_] := Length[Select[Range[10^n], PrimeQ[#] && !PrimeQ[2#+1] && !PrimeQ[4#+1]&]]; Table[f[n], {n,7}]

a(n) = A006880(n) - A092816(n) - A182265(n) + 1.

a(n) ~ 10^n / (n log 10). - Charles R Greathouse IV, May 11 2012

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A023212 Primes p such that 4*p+1 is also prime.

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A092816 Number of Sophie Germain primes less than 10^n.

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A182434 Number of primes p < n such that 4*p+1 is also prime.

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A210684 Number of primes p < 10^n such that both 2*p+1 and 4*p+1 are composite.

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A210684 Number of primes p < 10^n such that both 2p+1 and 4p+1 are composite.