A005385 -id:A005385 - cp's OEIS frontend

A195324 Even numbers that cannot be represented as the sum of an odd prime and a safe prime (A005385).

2, 4, 6, 32, 56, 92, 98, 122, 128, 140, 152, 176, 194, 212, 224, 242, 254, 260, 272, 296, 302, 308, 326, 332, 368, 392, 398, 410, 422, 434, 452, 458, 476, 488, 500, 512, 518, 524, 536, 542, 560, 572, 596, 602, 632, 644, 656, 662, 674, 686, 692, 704, 710, 728

Offset: 1

Dan Brumleve, Sep 15 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
StackExchange, Even numbers greater than 6 as sum of two specific primes

Cf. A000040 (primes), A005385 (safe primes).

a195324 n = a195324_list !! (n-1)
a195324_list = filter p [2,4..] where
   p n = all ((== 0) . a010051) $ takeWhile (> 1) $ map (n -) a005385_list
-- Reinhard Zumkeller, Sep 18 2011

mx = 1000; safe = Select[Prime[Range[PrimePi[mx]]], PrimeQ[(# - 1)/2] &]; p = Prime[Range[2, PrimePi[mx]]]; Complement[Range[2, mx, 2], Union[Flatten[Outer[Plus, p, t]]]] (* T. D. Noe, Sep 15 2011 *)

A227853 Numbers k such that 4620*k - 1 is a safe prime (A005385).

Original entry on oeis.org

5, 9, 10, 20, 24, 39, 49, 50, 58, 67, 79, 88, 91, 97, 98, 103, 116, 117, 134, 136, 137, 143, 148, 149, 180, 181, 182, 193, 201, 213, 230, 234, 239, 247, 253, 261, 265, 267, 275, 284, 285, 286, 288, 296, 301, 304, 313, 321, 325, 339, 347

Offset: 1

M. F. Hasler, Oct 31 2013

nonn

The given expression seems to produce safe primes quite frequently. The numbers 6*(4620*a(n)-1) form a subsequence of A230032.
Note that 4620 = 2^2*3*5*7*11.
Is a(4135..4139) = 59497..59501 the only set of 5 terms with common difference = 1? Checked for all terms up to 10^7. - Zak Seidov, Nov 02 2013
This is common because 4620 = 12*5*7*11 contains several small prime factors, which reduces the possible 12x-1 remainders, forcing all safe primes to be in other 12x-1 remainders mod 4620. Something similar would happen if 4620 were replaced with 12*5*7*13 = 5460. - Mark Andreas, Dec 31 2021

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A005385, A230032.

{f=4*3*5*7*11; for(k=1,2000,isprime(p=k*f-1)&&isprime(p\2)&&print1(k","))}

A284660 Terms of A161897 that are not in A005385.

Original entry on oeis.org

3083, 4931, 6563, 9923, 166667, 865643, 1306667, 2266883, 3367367, 3906563, 4128959, 5493179, 5591039, 6040187, 9122963, 9402179, 9871403, 10174343, 13081379, 13756403, 14924003, 16550243, 24165287, 29492747, 32140859, 34633427, 38425643, 42249587, 42258779, 43014659, 45067523, 52678643

Offset: 1

Robert Israel, Mar 31 2017

nonn

Primes p such that q = (p-1)/2 is composite, 3^q == 1 (mod p) and 3^(q-1) == 1 (mod p-1).

All terms == 5 (mod 6).

			p = 3083 is in the sequence because it is prime, q = (3083-1)/2 = 23*67 is composite, 3^q == 1 (mod p) and 3^(q-1) == 1 mod (p-1).

Robert Israel, Table of n, a(n) for n = 1..212

Cf. A005385, A161897.

filter:= p -> isprime(p) and not isprime((p-1)/2) and
   3&^((p-3)/2)  mod (p-1) = 1 and
   3 &^((p-1)/2) mod p = 1;
select(filter, [seq(p, p=5..10^7, 6)]); # Robert Israel, Mar 31 2017

A066868 a(n) is the least k such that n + Sum_{i=1..k} A005385(i) is prime; or 0 if none exists.

Original entry on oeis.org

2, 1, 10, 5, 2, 1, 2, 1, 12, 5, 2, 1, 4, 1, 4, 5, 2, 1, 2, 3, 4, 7, 12, 1, 2, 1, 4, 7, 2, 3, 2, 1, 4, 5, 2, 1, 4, 1, 6, 17, 2, 1, 4, 3, 6, 5, 2, 1, 2, 3, 4, 11, 12, 1, 2, 1, 4, 5, 2, 3, 2, 1, 4, 5, 14, 1, 2, 1, 10, 5, 2, 7, 10, 1, 6, 7, 2, 1, 8, 3, 4, 7, 26, 1

Offset: 1

Joseph L. Pe, Jan 21 2002

Conjecture: a(n) is nonzero for all n, so every n can be represented as the difference between a prime and a partial sum of the safe primes series. See A066753 for a similar conjecture.

			4 + (5 + 7 + 11 + 23 + 47) = 97, a prime and 5 consecutive safe primes, starting from the first safe prime 5, are needed to achieve this. Hence a(4) = 5.

Cf. A005385, A066753.

a(n) = my(p=3, s=n); for(k=1, oo, until(isprime((p-1)/2), p=nextprime(p+1)); if(isprime(s+=p), return(k))); \\ Jinyuan Wang, Jul 30 2020

Offset changed to 1 by Jinyuan Wang, Jul 30 2020

A075707 Safe primes (A005385) (p and (p-1)/2 are primes) such that 12*p+1 is also prime.

Original entry on oeis.org

5, 23, 59, 83, 383, 479, 503, 719, 839, 863, 1619, 2039, 2099, 2579, 2819, 2879, 3023, 4139, 4259, 4679, 4703, 4919, 5879, 6719, 6779, 7559, 8039, 8783, 8819, 10799, 11279, 11423, 12203, 12659, 12899, 12983, 13523, 13799, 14159, 14303, 14699, 15683, 18119, 18443, 19259, 19379, 20183, 20663, 21059, 23663, 24083, 24239, 24659, 27239, 28163, 29123, 29339, 29483, 29759, 30803, 31139, 31583, 36923, 37463, 38603, 39119, 39503, 39839, 39983, 41879, 42299, 42443, 43403, 44519, 44939, 46679, 47339, 47819, 47963

Offset: 1

Jani Melik, Oct 02 2002

nonn

			23 is a prime, so is (23-1)/2=11 and also 12*23+1=277, 59 is a prime, (59-1)/2=29 and 12*59+1=709, ...

Cf. A005384, A005385, A059455.

ts_sgB_var_pras := proc(nmax) local i,tren,atek; tren := 0: for i from 1 to nmax do atek := numtheory[safeprime](i): if (atek > tren) then if (isprime(atek)='true' and isprime(6*atek+1)='true') then tren := atek: fi; fi; od; end: seq(ts_sgB_var_pras(i), i=1..3000);

okQ[n_]:=PrimeQ[(n-1)/2]&&PrimeQ[12n+1]
Select[Prime[Range[5000]],okQ] (* Harvey P. Dale, Nov 21 2010 *)

More terms from Harvey P. Dale, Nov 21 2010

A347226 Safe primes (A005385) that are half-period primes (A097443).

Original entry on oeis.org

83, 107, 227, 347, 359, 467, 479, 563, 587, 719, 839, 1187, 1283, 1307, 1319, 1439, 1523, 1907, 2027, 2039, 2879, 2963, 2999, 3119, 3203, 3467, 3803, 3947, 4079, 4283, 4547, 4679, 4787, 4799, 4919, 5387, 5399, 5483, 5507, 5639, 5879, 6599, 6719, 6827, 7079, 7187, 7523

Offset: 1

Lamine Ngom, Aug 24 2021

Apart from 5 and 11, a safe prime p is necessarily either a full reptend prime (A001913) or a half-period prime (A097443) since (p-1) is semiprime: 2*A005384(n) (Sophie Germain primes).
Safe primes being full reptend primes are listed in A000353.
a(n) is of the form 100*k + 10*{0, 2, 4, 6, 8} + {3, 7} or 100*k + 10*{1, 3, 5, 7, 9} + 9.
Number of terms < 10^k: 0, 1, 11, 56, 343, 2138, 15267, 114847, 886907, 7079602, ...

			(107-1)/2 = 53 is a prime, and the periodic part of the decimal expansion of 1/107 is of length 53.
Hence the safe prime 107 is in the sequence.

Cf. A005385, A097443, A005384, A000353, A001913, A002371, A006883.

select(t -> isprime(t) and isprime((t - 1)/2) and numtheory:-order(10, t) = (t - 1)/2, [seq(t, t = 3 .. 10000, 2)]);

Select[Prime@Range@1000,PrimeQ[(#-1)/2]&&Length[First@@RealDigits[1/#]]==(#-1)/2&] (* Giorgos Kalogeropoulos, Sep 14 2021 *)

A005385 INTERSECTION A097443.
a(n) == {17, 23, 29} mod 30.
a(n) == 11 (mod 12). - Hugo Pfoertner, Aug 24 2021

A373063 Greatest k >= 1 such that (p + 1 - 2^i) / 2^i is prime for i = 1..k and p is prime from A005385.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2

Offset: 1

Ctibor O. Zizka, May 21 2024

nonn

"k-safe primes" iff (p + 1 - 2^i) / 2^i is prime for i = 0..k, p a prime number. A000040 are 0-safe primes, A005385 are 1-safe primes, A066179 are 2-safe primes.

			p = 11: (11 + 1 - 2^i) / 2^i is prime for i = 1..2, thus a(3) = 2.
p = 47: (47 + 1 - 2^i) / 2^i is prime for i = 1..4, thus a(5) = 4.

Cf. A000040, A005385, A066179.

isp(k) = (denominator(k) == 1) && isprime(k);
f(p) = my(i=1); while (isp((p+1-2^i)/2^i), i++); i-1;
apply(f, select(x->isp((x-1)/2), primes(1000))) \\ Michel Marcus, May 28 2024

A005384 Sophie Germain primes p: 2p+1 is also prime.

Original entry on oeis.org

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559

Offset: 1

N. J. A. Sloane

Then 2p+1 is called a safe prime: see A005385.
Primes p such that the equation phi(x) = 2p has solutions, where phi is the totient function. See A087634 for another such collection of primes. - T. D. Noe, Oct 24 2003
Subsequence of A117360. - Reinhard Zumkeller, Mar 10 2006
Let q = 2n+1. For these n (and q), the difference of two cyclotomic polynomials can be written as a cyclotomic polynomial in x^2: Phi(q,x) - Phi(2q,x) = 2x Phi(n,x^2). - T. D. Noe, Jan 04 2008
A Sophie Germain prime p is 2, 3 or of the form 6k-1, k >= 1, i.e., p = 5 (mod 6). A prime p of the form 6k+1, k >= 1, i.e., p = 1 (mod 6), cannot be a Sophie Germain prime since 2p+1 is divisible by 3. - Daniel Forgues, Jul 31 2009
Also solutions to the equation: floor(4/A000005(2*n^2+n)) = 1. - Enrique Pérez Herrero, May 03 2012
In the spirit of the conjecture related to A217788, we conjecture that for any integers n >= m > 0 there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) is prime. - Zhi-Wei Sun, Mar 26 2013
If k is the product of a Sophie Germain prime p and its corresponding safe prime 2p+1, then a(n) = (k-phi(k))/3, where phi is Euler's totient function. - Wesley Ivan Hurt, Oct 03 2013
Giovanni Resta found the first Sophie Germain prime which is also a Brazilian number (A125134), 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73. - _Bernard Schott, Mar 07 2019
For all Sophie Germain primes p >= 5, 2*p + 1 = min(A, B) where A is the smallest prime factor of 2^p - 1 and B the smallest prime factor of (2^p + 1) / 3. - Alain Rocchelli, Feb 01 2023
Consider a pair of numbers (p, 2*p+1), with p >= 3. Then p is a Sophie Germain prime iff (p-1)!^2 + 6*p == 1 (mod p*(2*p+1)). - Davide Rotondo, May 02 2024

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Peretti, The quantity of Sophie Germain primes less than x, Bull. Number Theory Related Topics, Vol. 11, No. 1-3 (1987), pp. 81-92.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 76, 227-230.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

J. S. Cheema, Table of n, a(n) for n = 1..100000. [This replaces an earlier b-file computed by T. D. Noe]
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
P. Bruillard, S.-H. Ng, E. Rowell and Z. Wang, On modular categories, arXiv preprint arXiv:1310.7050 [math.QA], 2013.
Chris K. Caldwell, The Prime Glossary, Sophie Germain Prime
Andrea Del Centina, Letters of Sophie Germain preserved in Florence Historia Mathematica, Vol. 32 (2005), pp. 60-75.
Harvey Dubner, Large Sophie Germain Primes, Math. Comp., Vol. 65, No. 213 (1996), pp. 393-396.
Luis H. Gallardo and Olivier Rahavandrainy, There are finitely many even perfect polynomials over F_p with p+1 irreducible divisors, Acta Mathematica Universitatis Comenianae, Vol. 83, No. 2, 2016, 261-275.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Reikku Kulon, Sublinear arbitrary precision generator of Sophie Germain and safe primes in C (public domain).
Henri Lifchitz, A new and simpler primality test for Sophie-Germain numbers(q=2*p+1).
Victor Meally, Letter to N. J. A. Sloane, no date.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
Frans Oort, Prime numbers, 2013.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 31, 114.
Larry Riddle, Sophie Germain and Fermat's Last Theorem, Agnes Scott College, Math. Dept., Jul, 1999.
Carlos Rivera, Puzzle 1122. OEIS A005385, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 1140. Test for Sophie Germain primes, The Prime Puzzles & Problems Connection.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Anthony G. Shannon, Peter J.-S. Shiue, Tian-Xiao He, and Christopher Saito, On the special cases of Carmichael's totient conjecture, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 504-534.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.
Agoh Takashi, On Sophie Germain primes, Number theory (Liptovský Ján, 1999), Tatra Mt. Math. Publ., Vol. 20 (2000), pp. 65-73.
Terence Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Vmoraru, PlanetMath.org, Sophie Germain prime.
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.
Eric Weisstein's World of Mathematics, Sophie Germain Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Sophie Germain prime.
Samuel Yates, Sophie Germain primes, in "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991, pp. 882-886.

Cf. A005385, A057331, A005602, A087634.
Cf. also A000355, A156541, A156542, A156592, A161896, A156660, A156874, A092816, A023212, A007528 (primes of the form 6k-1).
For primes p that remains prime through k iterations of the function f(x) = 2x + 1: this sequence (k=1), A007700 (k=2), A023272 (k=3), A023302 (k=4), A023330 (k=5), A278932 (k=6), A138025 (k=7), A138030 (k=8).

Filtered([1..1600],p->IsPrime(p) and IsPrime(2*p+1)); # Muniru A Asiru, Mar 06 2019

[ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; // Klaus Brockhaus, Jan 01 2009

A:={}: for n from 1 to 246 do if isprime(2*ithprime(n)+1) then A:=A union {ithprime(n)} fi od: A:=A; # Emeric Deutsch, Dec 09 2004

Select[Prime[Range[1000]],PrimeQ[2#+1]&]
lst = {}; Do[If[PrimeQ[n + 1] && PrimeOmega[n] == 2, AppendTo[lst, n/2]], {n, 2, 10^4}]; lst (* Hilko Koning, Aug 17 2021 *)

select(p->isprime(2*p+1), primes(1000)) \\ In old PARI versions <= 2.4.2, use select(primes(1000), p->isprime(2*p+1)).

forprime(n=2, 10^3, if(ispseudoprime(2*n+1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

is_A005384=(p->isprime(2*p+1)&&isprime(p));
  {A005384_vec(N=100,p=1)=vector(N,i,until(isprime(2*p+1),p=nextprime(p+1));p)} \\ M. F. Hasler, Mar 03 2020

from sympy import isprime, nextprime
def ok(p): return isprime(2*p+1)
def aupto(limit): # only test primes
  alst, p = [], 2
  while p <= limit:
    if ok(p): alst.append(p)
    p = nextprime(p)
  return alst
print(aupto(1559)) # Michael S. Branicky, Feb 03 2021

a(n) mod 10 <> 7. - Reinhard Zumkeller, Feb 12 2009
A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= n. - Reinhard Zumkeller, Feb 18 2009
tau(4*a(n) + 2) = tau(4*a(n)) - 2, for n > 1. - Arkadiusz Wesolowski, Aug 25 2012
eulerphi(4*a(n) + 2) = eulerphi(4*a(n)) + 2, for n > 1. - Arkadiusz Wesolowski, Aug 26 2012
A005097 INTERSECT A000040. - R. J. Mathar, Mar 23 2017
Sum_{n>=1} 1/a(n) is in the interval (1.533944198, 1.8026367) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021
a(n) >> n log^2 n. - Charles R Greathouse IV, Jul 25 2024

A005383 Primes p such that (p+1)/2 is prime.

Original entry on oeis.org

3, 5, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253

Offset: 1

N. J. A. Sloane

Also, n such that sigma(n)/2 is prime. - Joseph L. Pe, Dec 10 2001; confirmed by Vladeta Jovovic, Dec 12 2002
Primes that are followed by twice a prime, i.e., are followed by a semiprime. (For primes followed by two semiprimes, see A036570.) - Zak Seidov, Aug 03 2013, Dec 31 2015
If A005382(n) is in A168421 then a(n) is a twin prime with a Ramanujan prime, A104272(k) = a(n) - 2. - John W. Nicholson, Jan 07 2016
Starting with 13 all terms are congruent to 1 mod 12. - Zak Seidov, Feb 16 2017
Numbers n such that both n and n+12 are terms are 61, 661, 1201, 4261, 5101, 6121, 6361 (all congruent to 1 mod 60). - Zak Seidov, Mar 16 2017
Primes p for which there exists a prime q < p such that 2q == 1 (mod p). Proof: q = (p + 1)/2. - David James Sycamore, Nov 10 2018
Prime numbers n such that phi(sigma(2n)) = phi(2n), excluding n=3 and n=5; as well as phi(sigma(3n)) = phi(3n), excluding n=3 only. - Richard R. Forberg, Dec 22 2020

			Both 3 and (3+1)/2 = 2 are primes, both 5 and (5+1)/2 = 3 are primes. - _Zak Seidov_, Nov 19 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Benoit Cloitre, On the fractal behavior of primes, 2011.

Cf. A005382, A057326, A057327, A057328, A057329, A057330, A005603.
A subsequence of A000040 which has A036570 as subsequence.
Cf. A005385, A010051, A065091, A048161, A036570.

a005383 n = a005383_list !! (n-1)
a005383_list = [p | p <- a065091_list, a010051 ((p + 1) `div` 2) == 1]
-- Reinhard Zumkeller, Nov 06 2012

LIMIT = 8000 % Find all members of A005383 less than LIMIT A = primes(LIMIT); n = length(A); %n is number of primes less than LIMIT B = 2*A - 1; C = ones(n, 1)*A; %C is an n X n matrix, with C(i, j) = j-th prime D = B'*ones(1, n); %D is an n X n matrix, with D(i, j) = (i-th prime)*2 - 1 [i, j] = find(C == D); A(j)

[n: n in [1..3300] | IsPrime(n) and IsPrime((n+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012

for n to 300 do
  X := ithprime(n);
Y := ithprime(n+1);
Z := 1/2 mod Y;
  if isprime(Z) then print(Y);
end if:
end do:
# David James Sycamore, Nov 11 2018

Select[Prime[Range[1000]], PrimeQ[(# + 1)/2] &] (* Zak Seidov, Nov 19 2012 *)

A005383_list(n) = select(m->isprime(m\2+1),primes(n)[2..n]) \\ Charles R Greathouse IV, Sep 25 2012

from sympy import isprime
[n for n in range(3, 5000) if isprime(n) and isprime((n + 1)//2)]
# Indranil Ghosh, Mar 17 2017

[n for n in prime_range(3, 1000) if is_prime((n + 1) // 2)]
# F. Chapoton, Dec 17 2019

a(n) = A129521(n)/A005382(n). - Reinhard Zumkeller, Apr 19 2007
A000035(a(n))*A010051(a(n))*A010051((a(n)+1)/2) = 1. - Reinhard Zumkeller, Nov 06 2012
a(n) = 2*A005382(n) - 1. - Zak Seidov, Nov 19 2012
a(n) = A005382(n) + phi(A005382(n)) = A005382(n) + A000010(A005382(n)). - Torlach Rush, Mar 10 2014

More terms from David Wasserman, Jan 18 2002

Name changed by Jianing Song, Nov 27 2021

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

Original entry on oeis.org

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617

Offset: 1

Clark Kimberling, Dec 15 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...).
See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x        Lower POBA       Upper POBA       POBA
1        A001359/A006512  A006512/A001359  A265759/A265760
3/2      A104163/A158708  A162336/A158709  A265761/A222565
2        A005383/A005382  A005385/A005384  A079149/A120628
3        A091180/A088878  A094525/A023208  A265763/A265764
4        A162857/A062737  A090866/A023212  A265765/A120639
5        A265766/A158318  A265767/A023217  A265768/A265769
6        A227756/A158015  A051644/A007693  A265770/A265771
sqrt(2)  A265772/A265773  A265774/A265775  A265776/A265777
sqrt(3)  A265778/A265779  A265780/A265781  A265782/A265783
sqrt(5)  A265784/A265785  A265786/A265787  A265788/A265789
sqrt(8)  A265790/A265791  A265792/A265793  A265794/A265795
tau      A265796/A265797  A265798/A265799  A265800/A265801
1/tau    A265799/A265798  A265797/A265796  A265806/A265807
pi       A265808/A265809  A265810/A265811  A265812/A265813
e        A265814/A265815  A265816/A265817  A265818/A265819

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Cf. A000040, A001359, A006512, A265759, A265760.

x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

cp's OEIS Frontend

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A347226 Safe primes (A005385) that are half-period primes (A097443).

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A373063 Greatest k >= 1 such that (p + 1 - 2^i) / 2^i is prime for i = 1..k and p is prime from A005385.

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A005384 Sophie Germain primes p: 2p+1 is also prime.

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