A005384 -id:A005384 - cp's OEIS frontend

A059455 Safe primes which are also Sophie Germain primes.

5, 11, 23, 83, 179, 359, 719, 1019, 1439, 2039, 2063, 2459, 2819, 2903, 2963, 3023, 3623, 3779, 3803, 3863, 4919, 5399, 5639, 6899, 6983, 7079, 7643, 7823, 10163, 10799, 10883, 11699, 12203, 12263, 12899, 14159, 14303, 14699, 15803, 17939

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Primes p such that both (p-1)/2 and 2*p+1 are prime.
Except for 5, all are congruent to 11 modulo 12.
Primes "inside" Cunningham chains of first kind.
Infinite under Dickson's conjecture. - Charles R Greathouse IV, Jul 18 2012
See A162019 for the subset of a(n) that are "reproduced" by the application of the transformations (a(n)-1)/2 and 2*a(n)+1 to the set a(n). - Richard R. Forberg, Mar 05 2015

			83 is a term because it is prime and 2*83+1 = 167 and (83-1)/2 = 41 are both primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A005385.

Cf. A053176, A059452, A059453, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660, A156877, A162019.

[p: p in PrimesUpTo(20000) |IsPrime((p-1) div 2) and IsPrime(2*p+1)]; // Vincenzo Librandi, Oct 31 2014

lst={}; Do[p=Prime[n]; If[PrimeQ[(p-1)/2]&&PrimeQ[2*p+1], AppendTo[lst, p]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 02 2008 *)
Select[Prime[Range[1000]], AllTrue[{(# - 1)/2, 2 # + 1}, PrimeQ] &] (* requires Mathematica 10+; Feras Awad, Dec 19 2018 *)

forprime(p=2,1e5,if(isprime(p\2)&&isprime(2*p+1),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

from itertools import count, islice
from sympy import isprime, prime
def A059455_gen(): # generator of terms
    return filter(lambda p:isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059455_list = list(islice(A059455_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156660(a(n))*A156659(a(n)) = 1; A156877 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

A023272 Primes that remain prime through 3 iterations of the function f(x) = 2*x + 1.

Original entry on oeis.org

2, 5, 89, 179, 359, 509, 1229, 1409, 2699, 3539, 6449, 10589, 11549, 11909, 12119, 17159, 19709, 19889, 22349, 26189, 27479, 30389, 43649, 53639, 53849, 55229, 57839, 60149, 61409, 63419, 66749, 71399, 74699, 75329, 82499, 87539, 98369, 101399, 104369

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+1, 4*p+3 and 8*p+7 are also primes. - Vincenzo Librandi, Aug 04 2010

For n > 2, a(n) == 29 (mod 30). - Zak Seidov, Jan 31 2013

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Intersection of A007700 and A023231.

Cf. A005384, A005385, A023302, A023330, A057331, A005602.

[n: n in [1..100000] | IsPrime(n) and IsPrime(2*n+1) and IsPrime(4*n+3) and IsPrime(8*n+7)] // Vincenzo Librandi, Aug 04 2010

p:=2: for n from 1 to 5000 do if(isprime(2*p+1) and isprime(4*p+3) and isprime(8*p+7))then printf("%d, ",p): fi: p:=nextprime(p): od: # Nathaniel Johnston, Jun 30 2011

Select[Prime[Range[10^3*4]], PrimeQ[a1=2*#+1] && PrimeQ[a2=2*a1+1] && PrimeQ[a3=2*a2+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Join[{2, 5}, Select[Range[89, 104369, 30], PrimeQ[#] && PrimeQ[2*# + 1] && PrimeQ[4*# + 3] && PrimeQ[8*# + 7] &]] (* Zak Seidov, Jan 31 2013 *)
p3iQ[n_]:=AllTrue[NestList[2#+1&,n,3],PrimeQ]; Join[{2,5},Select[ Range[ 89,200000,30],p3iQ]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 30 2019 *)

is(n)=isprime(n)&&isprime(2*n+1)&&isprime(4*n+3)&&isprime(8*n+7) \\ Charles R Greathouse IV, Mar 21 2013

A023330 Primes that remain prime through 5 iterations of function f(x) = 2x + 1.

Original entry on oeis.org

89, 63419, 127139, 405269, 810809, 1069199, 1122659, 1178609, 1333889, 1598699, 1806089, 1958249, 2164229, 2245319, 2329469, 2606069, 2848949, 3241289, 3339989, 3784199, 3962039, 4088879, 4328459, 4444829, 4658939, 4664249, 4894889, 4897709, 5132999

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+1, 4*p+3, 8*p+7, 16*p+15 and 32*p+31 are also primes. - Vincenzo Librandi, Aug 04 2010

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A007700, A023272, A023302, A057331, A005602.

[n: n in [1..5000000] | forall{2^i*n+2^i-1: i in [0..5] | IsPrime(2^i*n+2^i-1)}]; // Vincenzo Librandi, Aug 04 2010

Select[Prime[Range[10^5]], PrimeQ[a1=2*#+1] && PrimeQ[a2=2*a1+1] && PrimeQ[a3=2*a2+1] && PrimeQ[a4=2*a3+1] && PrimeQ[a5=2*a4+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

is(n)=isprime(n) && isprime(2*n+1) && isprime(4*n+3) && isprime(8*n+7) && isprime(16*n+15) && isprime(32*n+31) \\ Charles R Greathouse IV, Jul 01 2013

from sympy import prime, isprime
A023330_list = [p for p in (prime(n) for n in range(1,10**5)) if all([isprime(2**m*(p+1)-1) for m in range(1,6)])] # Chai Wah Wu, Sep 09 2014

a(n) == 29 (mod 30). - Zak Seidov, Jan 31 2013

A023302 Primes that remain prime through 4 iterations of function f(x) = 2x + 1.

Original entry on oeis.org

2, 89, 179, 53639, 53849, 61409, 63419, 66749, 126839, 127139, 143609, 167729, 186149, 206369, 254279, 268049, 296099, 340919, 405269, 422069, 446609, 539009, 594449, 607319, 658349, 671249, 725009, 775949, 810539, 810809, 812849, 819509

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+1, 4*p+3, 8*p+7 and 16*p+15 are also primes. - Vincenzo Librandi, Aug 04 2010

For n > 1, a(n) == 29 (mod 30). One should use it in codes. - Zak Seidov, Jan 31 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A007700, A023272, A023330, A057331, A005602.

[n: n in [1..1200000] | IsPrime(n) and IsPrime(2*n+1) and IsPrime(4*n+3) and IsPrime(8*n+7) and IsPrime(16*n+15)] // Vincenzo Librandi, Aug 04 2010

Select[Prime[Range[10^4*4]], PrimeQ[a1=2*#+1] && PrimeQ[a2=2*a1+1] && PrimeQ[a3=2*a2+1] && PrimeQ[a4=2*a3+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Join[{2},Select[Range[29,820000,30],And@@PrimeQ[NestList[2#+1&,#,4]]&]] (* Harvey P. Dale, Apr 03 2013 *)

is(n)=isprime(n) && isprime(2*n+1) && isprime(4*n+3) && isprime(8*n+7) && isprime(16*n+15) \\ Charles R Greathouse IV, Jul 01 2013

A008347 a(n) = Sum_{i=0..n-1} (-1)^i * prime(n-i).

Original entry on oeis.org

0, 2, 1, 4, 3, 8, 5, 12, 7, 16, 13, 18, 19, 22, 21, 26, 27, 32, 29, 38, 33, 40, 39, 44, 45, 52, 49, 54, 53, 56, 57, 70, 61, 76, 63, 86, 65, 92, 71, 96, 77, 102, 79, 112, 81, 116, 83, 128, 95, 132, 97, 136, 103, 138, 113, 144, 119, 150, 121, 156, 125, 158, 135, 172, 139, 174, 143

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

Define the sequence b(n) by b(1) = 1; b(n) = 1 - (prime(n-1)/prime(n))*b(n-1) if n > 1. Then b(n) = a(n)/prime(n). Does lim b(n) exist? If so, it must equal 1/2. - Joseph L. Pe, Feb 17 2003
This sequence contains no duplicate values; after the initial 0, 2, the parity alternates, and a(n+2) > a(n). Do even and odd values trade the lead infinitely often (as would be expected if we model their difference as a random walk)? - Franklin T. Adams-Watters, Jan 25 2010
Conjecture: For any m = 1, 2, 3, ... and r = 0, ..., m - 1, there are infinitely many positive integers n with a(n) == r (mod m). - Zhi-Wei Sun, Feb 27 2013
From Zhi-Wei Sun, May 18 2013: (Start)
Conjectures:
(i) The sequence a(1), a(2), a(3), ... contains infinitely many Sophie Germain primes (A005384). (For example, a(1) = 2, a(4) = 3, a(6) = 5, a(18) = 29, a(28) = 53, a(46) = 83, a(54) = 113 and a(86) = 191 are Sophie Germain primes.) Also, there are infinitely many positive integers n such that a(n) - 1 and a(n) + 1 are twin primes. (Such integers n are 3, 7, 11, 41, 53, 57, 69, 95, 147, 191, 253, ....)
(ii) For each non-constant integer-valued polynomial P(x) with positive leading coefficient, there are infinitely many positive integers n such that a(n) = P(x) for some positive integer x. (For example, a(2) = 1^2, a(3) = 2^2, a(9) = 4^2, a(26) = 7^2, a(44) = 9^2, a(55) = 12^2 and a(58) = 11^2 are squares.)
(iii) The only powers of two in the current sequence are a(1) = 2, a(2) = 1, a(3) = 4, a(5) = 8, a(9) = 16, a(17) = 32, a(47) = 128, and a(165) = 512.
(iv) The only solutions to the equation a(n) = m! are (m,n) = (1, 2), (2, 1), (8, 7843). [False!] (End)
Conjecture: For any n > 9 we have a(n+1) < a(n-1)^(1+2/(n+2)). (This yields an upper bound for prime(n+1) - prime(n) in terms of prime(1), ..., prime(n-1). The conjecture has been verified for n up to 10^8.) - Zhi-Wei Sun, Jun 09 2013
Conjecture (iv) above is false since a(1379694977463) = 20922789888000 = 16!. - Giovanni Resta, Sep 04 2018
Conjecture: We have {a(m)+a(n): m,n>0} = {2,3,...}. Also, {a(m)-a(n): m,n>0} contains all the integers, and {a(m)/a(n): m,n>0} contains all the positive rational numbers. (I have noted that {a(m)/a(n): m,n = 1..60000} contains {a/b: a,b = 1..1000}.) - Zhi-Wei Sun, May 23 2019
Let d(n) = a(n) - a(n-1). Since a(n-1) = prime(n) - a(n), d(n) = 2*a(n) - prime(n). If lim inf a(n)/prime(n) = 1/2 as conjectured by Joseph L. Pe above holds, lim inf d(n)/prime(n) = 2*lim inf a(n)/prime(n) - 1 = 0. Numerical analysis of a(n) for n up to 10^9 shows that abs(d(n))/sqrt(prime(n)) < 15, and thus abs(d(n)) = O(sqrt(prime(n))) is conjectured. - Ya-Ping Lu, Aug 31 2020

N. J. A. Sloane, Table of n, a(n) for n = 0..30000 (updated Dec 18 2019; terms 0..2000 from T. D. Noe, terms 2001..10000 from Robert G. Wilson v)
Romeo Meštrovic, On the distribution of primes in the alternating sums of consecutive primes, arXiv:1805.11657 [math.NT], 2018.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.
Zhi-Wei Sun, On a sequence involving sums of primes, Bull. Aust. Math. Soc. 88(2013), 197-205.

Complement is in A226913.

Cf. A000040, A001223, A005384, A007504, A181901.

a008347 n = a008347_list !! n
a008347_list = 0 : zipWith (-) a000040_list a008347_list
-- Reinhard Zumkeller, Feb 09 2015

[0] cat [&+[ (-1)^k * NthPrime(n-k): k in [0..n-1]]: n in [1..70]]; // Vincenzo Librandi, May 26 2019

A008347 := proc(n) options remember; if n = 0 then 0 else abs(A008347(n-1)-ithprime(n)); fi; end;

Join[{0},Abs[Accumulate[Times@@@Partition[Riffle[Prime[Range[80]],{1,-1}], 2]]]] (* Harvey P. Dale, Dec 11 2011 *)
f[n_] := Abs@ Sum[(-1)^k Prime[k], {k, n - 1}]; Array[f, 70] (* Robert G. Wilson v, Oct 08 2013 *)
a[0] = 0; a[n_] := a[n] = Prime[n] - a[n - 1]; Array[a, 70, 0] (* Robert G. Wilson v, Oct 16 2013 *)
FoldList[#2 - # &, 0, Array[Prime, 30]] (* Horst H. Manninger, Oct 29 2021 *)

a(n)=abs(sum(i=1,n,(-1)^i*prime(i))) \\ Charles R Greathouse IV, Apr 29 2015

from sympy import nextprime
p = a = 0; L = [a]
for n in range(1, 67): p = nextprime(p); a = p - a; L.append(a)
print(*L, sep = ", ") # Ya-Ping Lu, May 07 2023

a(n) = prime(n) - a(n-1) for n >= 1.
a(n+2) - a(n) = A001223(n+1). - Reinhard Zumkeller, Feb 09 2015
G.f: (x*b(x))/(1+x), where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 10 2016
Meštrovic (2018), following Pillai, conjectures that
  a(2k) = k*log k + k*loglog k - k + o(k) as k -> oo,
with a similar conjecture for a(2k+1). - N. J. A. Sloane, Dec 21 2019

A111153 Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime.

Original entry on oeis.org

4, 10, 25, 34, 38, 46, 55, 57, 77, 91, 93, 106, 118, 123, 129, 133, 143, 145, 159, 161, 169, 177, 185, 201, 203, 205, 206, 213, 218, 226, 235, 259, 267, 289, 291, 295, 298, 305, 314, 327, 334, 335, 339, 358, 361, 365, 377, 381, 394, 395, 403, 407, 415, 417

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Oct 19 2005

nonn

Define a generalized Sophie Germain n-prime of degree m, p, to be an n-prime (n-almost prime) such that 2p+1 is an m-prime (m-almost prime). For example, p=24 is a Sophie Germain 4-prime of degree 2 because 24 is a 4-prime and 2*24+1=49 is a 2-prime. Then this sequence gives all the Sophie Germain 2-primes of degree 2.

			a(4)=34 because 34 is the 4th semiprime such that 2*34+1=69 is also a semiprime.

Marius A. Burtea, Table of n, a(n) for n = 1..7675 (first 1000 terms from T. D. Noe)

Cf. A005384, A001358, A111168, A111170, A111171, A111173, A111176, A176896.

f:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [4..500] | f(n) and f(2*n+1)]; // Marius A. Burtea, Jan 04 2019

SemiPrimeQ[n_] := (Plus@@Transpose[FactorInteger[n]][[2]]==2); Select[Range[2, 500], SemiPrimeQ[ # ]&&SemiPrimeQ[2#+1]&] (* T. D. Noe, Oct 20 2005 *)
fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range[445], fQ[ # ] && fQ[2# + 1] &] (* Robert G. Wilson v, Oct 20 2005 *)
Flatten@Position[PrimeOmega@{#,1+2*#}&/@Range@1000,{2,2}] (* Hans Rudolf Widmer, Nov 25 2023 *)

isok(n) = (bigomega(n) == 2) && (bigomega(2*n+1) == 2); \\ Michel Marcus, Jan 04 2019

a(n) = (A176896(n) - 1)/2. - Zak Seidov, Sep 10 2012

Corrected and extended by T. D. Noe, Ray Chandler and Robert G. Wilson v, Oct 20 2005

A059452 Safe primes (A005385) that are not Sophie Germain primes.

Original entry on oeis.org

7, 47, 59, 107, 167, 227, 263, 347, 383, 467, 479, 503, 563, 587, 839, 863, 887, 983, 1187, 1283, 1307, 1319, 1367, 1487, 1523, 1619, 1823, 1907, 2027, 2099, 2207, 2447, 2579, 2879, 2999, 3119, 3167, 3203, 3467, 3947, 4007, 4079, 4127, 4139, 4259, 4283

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Except for 7, these primes are congruent to 11 modulo 12.

Terminal primes in complete Cunningham chains of first kind, i.e., the chains cannot be continued from these primes.

			347 is a term because 173 is a prime but 695 is not.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chain.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains. [Wayback Machine link]

Intersection of A005385 and A053176.

Cf. A005384, A059452, A059453, A059454, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660.

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)/2],If[ !PrimeQ[2*p+1],AppendTo[lst,p]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 24 2009 *)

is(p) = p > 2 && isprime(p) && isprime((p-1)/2) && !isprime(2*p+1); \\ Amiram Eldar, Jul 15 2024

from itertools import count, islice
from sympy import isprime, prime
def A059452_gen(): # generator of terms
    return filter(lambda p:isprime(p>>1) and not isprime(p<<1|1),(prime(i) for i in count(1)))
A059452_list = list(islice(A059452_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156659(a(n))*(1-A156660(a(n))) = 1. - Reinhard Zumkeller, Feb 18 2009

Broken link updated by R. J. Mathar, Apr 12 2010

A192580 Monotonic ordering of set S generated by these rules: if x and y are in S and xy+1 is a prime, then xy+1 is in S, and 2 is in S.

Original entry on oeis.org

2, 5, 11, 23, 47

Offset: 1

Clark Kimberling, Jul 04 2011

Following the discussion at A192476, the present sequence introduces a restriction: that the generated terms must be prime. A192580 is the first of an ascending chain of finite sequences, determined by the initial set called "start":
A192580:  f(x,y)=xy+1 and start={2}
A192581:  f(x,y)=xy+1 and start={2,4}
A192582:  f(x,y)=xy+1 and start={2,4,6}
A192583:  f(x,y)=xy+1 and start={2,4,6,8}
A192584:  f(x,y)=xy+1 and start={2,4,6,8,10}
For other choices of the function f(x,y) and start, see A192585-A192598.
A192580 consists of only 5 terms, A192581 of 7 terms, and A192582 of 28,...; what can be said about the sequence (5,7,28,...)?
2, 5, 11, 23, 47 is the complete Cunningham chain that begins with 2. Each term except the last is a Sophie Germain prime A005384. - Jonathan Sondow, Oct 28 2015

			2 is in the sequence by decree.
The generated numbers are 5=2*2+1, 11=2*5+1, 23=2*11+1, 47=2*23+1.

Wikipedia, Cunningham chain

Cf. A005384, A192476, A192581, A192582, A192583, A192584.

start = {2}; primes = Table[Prime[n], {n, 1, 10000}];
f[x_, y_] := If[MemberQ[primes, x*y + 1], x*y + 1]
b[x_] := Block[{w = x}, Select[Union[Flatten[AppendTo[w, Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # < 50000 &]];
t = FixedPoint[b, start]  (* A192580 *)

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

Original entry on oeis.org

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

A179882 a(n) is the corresponding value of contraharmonic mean B(h) of numbers k such that gcd(k, h) = 1 (k < h) for numbers h from A179877(n) and A179878(n).

Original entry on oeis.org

1, 7, 15, 31, 39, 55, 71, 111, 119, 151, 175, 177, 231, 239, 255, 303, 311, 313, 319, 329, 335, 337, 345, 375, 391, 393, 479, 521, 559, 575, 591, 593, 601, 607, 623, 655, 657, 679, 777, 785, 791, 823, 855, 863, 871, 879, 889, 905, 911, 929, 937, 959, 961, 991

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Subsequence of A179873 and A179874.
It appears that for n >= 3, (4*A005384(n)+1)/3 is a subsequence. - Hilko Koning, Jul 27 2018
This happens for this subsequence of A179877: 10, 22, 46, 58, 82, 106, 166, 178, ... apparently "Semiprimes of form prime - 1" >= 10 (see A077065). - Michel Marcus, Jul 27 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Contraharmonic mean.

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

{1}~Join~Select[Partition[Table[ContraharmonicMean@ Select[Range[n - 1], GCD[#, n] == 1 &], {n, 2, 1500}], 2, 1], And[IntegerQ@ First@ #, SameQ @@ #] &][[All, 1]] (* Michael De Vlieger, Jul 30 2018 *)

lista(nn) = {vch = vector(nn, k, ch(k)); for (i=1, nn-1, if ((vch[i] == vch[i+1]) && !frac(vch[i]), print1(vch[i], ", ")););} \\ Michel Marcus, Jul 27 2018

a(n) = A175505(A179877(n)) / A175506(A179877(n)).

a(n) = A175505(A179878(n)) / A175506(A179878(n)).

More terms from Michel Marcus, Jul 27 2018

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A059455 Safe primes which are also Sophie Germain primes.

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A023272 Primes that remain prime through 3 iterations of the function f(x) = 2*x + 1.

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A023330 Primes that remain prime through 5 iterations of function f(x) = 2x + 1.

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A023302 Primes that remain prime through 4 iterations of function f(x) = 2x + 1.

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A008347 a(n) = Sum_{i=0..n-1} (-1)^i * prime(n-i).

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A111153 Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime.

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