A005384 -id:A005384 - cp's OEIS frontend

A111206 Semi-Sophie Germain semiprimes: semiprimes which are the product of Sophie Germain primes.

4, 6, 9, 10, 15, 22, 25, 33, 46, 55, 58, 69, 82, 87, 106, 115, 121, 123, 145, 159, 166, 178, 205, 226, 249, 253, 262, 265, 267, 319, 339, 346, 358, 382, 393, 415, 445, 451, 466, 478, 502, 519, 529, 537, 562, 565, 573, 583, 586, 655, 667, 699, 717, 718, 753, 838

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Oct 24 2005

nonn

Define an n-almost Sophie Germain almost-prime to be an n-almost prime all the prime factors of which are Sophie Germain primes. Note the contrast between this terminology and that of Sophie Germain n-almost primes, they are different.

			a(4) = 10 because 10 is the 4th semiprime both the prime factors of which are Sophie Germain primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A111153, A001358, A005384, A111207.

lst={};Do[If[Plus@@Last/@FactorInteger[n]==2,a=First/@FactorInteger[n];b=a[[1]];k=0;If[Length[a]==2,c=a[[2]];If[ !PrimeQ[2*c+1],k=1]];If[PrimeQ[2*b+1]&&k==0,AppendTo[lst,n]]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 27 2009 *)
Module[{nn=100,sgp},sgp=Select[Prime[Range[100]],PrimeQ[2#+1]&];Select[ Union[ Times@@@Tuples[sgp,2]],#<=10nn&]] (* Harvey P. Dale, May 08 2019 *)

list(lim)=my(v=List(),u=v,t); forprime(p=2,lim\2, if(isprime(2*p+1), listput(u,p))); for(i=1,#u, for(j=1,i, t=u[i]*u[j]; if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Feb 05 2017

Extended by Ray Chandler, Oct 31 2005

A124485 Numbers n such that 2n-1 and 4n-1 are primes.

Original entry on oeis.org

2, 3, 6, 12, 15, 21, 27, 42, 45, 57, 66, 87, 90, 96, 117, 120, 126, 141, 147, 180, 210, 216, 222, 246, 255, 297, 321, 327, 330, 342, 360, 372, 381, 405, 456, 477, 507, 510, 516, 525, 552, 612, 615, 645, 705, 720, 726, 741, 750, 756, 780, 792, 801, 867, 906, 945

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A005384 (Sophie Germain primes).

Cf. A124486, A124487, A124488, A124489, A124490, A124491, A124492.

Filtered([1..1000],p->IsPrime(2*p-1) and IsPrime(4*p-1)); # Muniru A Asiru, Jul 19 2018

select(k->isprime(2*k-1) and isprime(4*k-1),[$1..1000]); # Muniru A Asiru, Jul 19 2018

Select[Range[1000], And @@ PrimeQ /@ ({2, 4}*# - 1) &] (* Ray Chandler, Nov 21 2006 *)

a(n) = (A005384(n+1) + 1)/2. - Hilko Koning, Jul 19 2018

Extended by Ray Chandler, Nov 21 2006

A209253 Number of ways to write 2n-1 as the sum of a Sophie Germain prime and a practical number.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 5, 2, 3, 4, 4, 4, 5, 2, 3, 5, 2, 4, 7, 4, 2, 6, 2, 5, 6, 2, 2, 6, 1, 3, 7, 4, 3, 7, 4, 5, 8, 2, 3, 8, 3, 3, 8, 4, 4, 7, 4, 5, 8, 3, 4, 7, 1, 5, 9, 5, 3, 9, 3, 4, 8, 4, 3, 9, 3, 5, 8, 2, 2, 9, 4, 3, 8, 4, 4, 10, 1, 3, 10, 5, 4, 10, 4, 3, 9, 5, 5, 10, 4

Offset: 1

Zhi-Wei Sun, Jan 14 2013

nonn

Conjecture: a(n)>0 for all n>1.

This has been verified for n up to 5*10^6.

			a(40)=1 since 2*40-1=23+56 with 23 a Sophie Germain prime and 56 a practical number.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 2012-2017

Cf. A005384, A005153, A208243, A208244, A208246, A208249.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n],s+1],1]<=DivisorSigma[1,Product[Pow[n,i],{i,1,s}]]+1,0,1],{s,1,Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[PrimeQ[2Prime[k]+1]==True&&pr[2n-1-Prime[k]]==True,1,0],{k,1,PrimePi[2n-1]}]
Do[Print[n," ",a[n]],{n,1,100}]

A002174 Values taken by reduced totient function psi(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 90, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168

Offset: 1

N. J. A. Sloane

If p is a Sophie Germain prime (A005384), then 2p is here. - T. D. Noe, Aug 13 2008
Terms of A002322, sorted and multiple values taken just once. - Vladimir Joseph Stephan Orlovsky, Jul 21 2009
a(2445343) = 10^7, suggesting that Luca & Pomerance's lower bound may be closer to the truth than the upper bound. The fit exponent log a(n)/log n - 1 = 0.0957... in this case. - Charles R Greathouse IV, Jul 02 2017

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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
D. H. Lehmer, Guide to Tables in the Theory of Numbers, Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
Florian Luca and Carl Pomerance, On the range of Carmichael's universal-exponent function, Acta Arithmetica 162 (2014), pp.289-308.
C. Moreau, Sur quelques théorèmes d'arithmétique, Nouvelles Annales de Mathématiques, 17 (1898), 293-307.

Cf. A002322, A002396, A143407, A143408.

lst={}; Do[AppendTo[lst, CarmichaelLambda[n]], {n, 6*7!}]; lst; Take[Union[lst], 123] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
(* warning: there seems to be no guarantee that no terms near the end are omitted! - Joerg Arndt, Dec 23 2014 *)
TakeWhile[Union@ Table[CarmichaelLambda@ n, {n, 10^6}], # <= 168 &] (* Michael De Vlieger, Mar 19 2016 *)

list(lim)=my(v=List([1]),u,t); forprime(p=3,lim\3+1, u=List(); listput(u,p-1); while((t=u[#u]*p)<=lim, listput(u,t)); for(j=1,#v, for(i=1,#u, t=lcm(u[i],v[j]); if(t<=lim && t!=v[j], listput(v,t)))); v=List(Set(v))); forprime(p=lim\3+2,lim+1, listput(v,p-1)); v=List(Set(v)); for(i=1,#v, t=2*v[i]; if(t>lim, break); listput(v,t); while((t*=2)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 23 2017

is(n)=if(n%2, return(n==1)); my(f=factor(n),pe); for(i=1,#f~, if(n%(f[i,1]-1)==0, next); pe=f[i,1]^f[i,2]; forstep(q=2*pe+1,n+1,2*pe, if(n%(q-1)==0 && isprime(q), next(2))); return(0)); 1 \\ Charles R Greathouse IV, Jun 25 2017

n (log n)^0.086 << a(n) << n (log n)^0.36 where << is the Vinogradov symbol, see Luca & Pomerance. - Charles R Greathouse IV, Dec 28 2013

More terms from T. D. Noe, Aug 13 2008

A059761 Initial primes of Cunningham chains of first type with length exactly 2. Primes in A059453 that survive as primes only one "2p-1 iteration", forming chains of exactly 2 terms.

Original entry on oeis.org

3, 29, 53, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1049, 1103, 1223, 1289, 1499, 1559, 1583, 1601, 1733, 1973, 2003, 2069, 2129, 2141, 2273, 2339, 2351, 2393, 2399, 2543

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3} = {composite, prime, prime, composite}.

			53 is a term because 26 and 215 are composites, and 53 and 107 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Chris Caldwell's Prime Glossary, Cunningham chains.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains. [Wayback Machine link]
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700.

ccftQ[p_]:=Boole[PrimeQ[{(p-1)/2,p,2 p+1,4 p+3}]]=={0,1,1,0}; Select[ Prime[ Range[400]],ccftQ] (* Harvey P. Dale, Jun 19 2021 *)

A054639 Queneau numbers: numbers n such that the Queneau-Daniel permutation {1, 2, 3, ..., n} -> {n, 1, n-1, 2, n-2, 3, ...} is of order n.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 14, 18, 23, 26, 29, 30, 33, 35, 39, 41, 50, 51, 53, 65, 69, 74, 81, 83, 86, 89, 90, 95, 98, 99, 105, 113, 119, 131, 134, 135, 146, 155, 158, 173, 174, 179, 183, 186, 189, 191, 194, 209, 210, 221, 230, 231, 233, 239

Offset: 1

Gilles Esposito-Farese (gef(AT)cpt.univ-mrs.fr), May 17 2000

nonn

The troubadour Arnaut Daniel composed sestinas based on the permutation 123456 -> 615243, which cycles after 6 iterations.
Roubaud quotes the number 141, but the corresponding Queneau-Daniel permutation is only of order 47 = 141/3.
This appears to coincide with the numbers n such that a type-2 optimal normal basis exists for GF(2^n) over GF(2). But are these two sequences really the same? - Joerg Arndt, Feb 11 2008
The answer is Yes - see Theorem 2 of the Dumas reference. [Jean-Guillaume Dumas (Jean-Guillaume.Dumas(AT)imag.fr), Mar 20 2008]
From Peter R. J. Asveld, Aug 17 2009: (Start)
a(n) is the n-th T-prime (Twist prime). For N >= 2, the family of twist permutations is defined by
p(m,N) == +2m (mod 2N+1) if 1 <= m < k = ceiling((N+1)/2),
p(m,N) == -2m (mod 2N+1) if k <= m < N.
N is T-prime if p(m,N) consists of a single cycle of length N.
The twist permutation is the inverse of the Queneau-Daniel permutation.
N is T-prime iff p=2N+1 is a prime number and exactly one of the following three conditions holds;
(1) N == 1 (mod 4) and +2 generates Z_p^* (the multiplicative group of Z_p) but -2 does not,
(2) N == 2 (mod 4) and both +2 and -2 generate Z_p^*,
(3) N == 3 (mod 4) and -2 generate Z_p^* but +2 does not. (End)
The sequence name says the permutation is of order n, but P. R. J. Asveld's comment says it's an n-cycle. Is there a proof that those conditions are equivalent for the Queneau-Daniel permutation? (They are not equivalent for any arbitrary permutation; e.g., (123)(45)(6) has order 6 but isn't a 6-cycle.) More generally, I have found that for all n <= 9450, (order of Queneau-Daniel permutation) = (length of orbit of 1) = A003558(n). Does this hold for all n? - David Wasserman, Aug 30 2011

			For N=6 and N=7 we obtain the permutations (1 2 4 5 3 6) and (1 2 4 7)(3 6)(5): 6 is T-prime, but 7 is not. - _Peter R. J. Asveld_, Aug 17 2009

Raymond Queneau, Note complémentaire sur la Sextaine, Subsidia Pataphysica 1 (1963), pp. 79-80.
Jacques Roubaud, Bibliothèque Oulipienne No 65 (1992) and 66 (1993).

P. R. J. Asveld, Table of n, a(n) for n = 1..10085
Jean-Paul Allouche, Manon Stipulanti, and Jia-Yan Yao, Doubling modulo odd integers, generalizations, and unexpected occurrences, arXiv:2504.17564 [math.NT], 2025.
Joerg Arndt, Matters Computational (The Fxtbook), section 42.9 "Gaussian normal bases", pp. 914-920
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers, TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some families of permutations and their primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Queneau Numbers--Recent Results and a Bibliography, University of Twente, 2013.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.
Michèle Audin, Poésie, Spirales, et Battements de Cartes, Images des Mathématiques, CNRS, 2019 (in French).
M. Bringer, Sur un problème de R. Queneau, Math. Sci. Humaines No. 25 (1969) 13-20.
Jean-Guillaume Dumas, Caractérisation des Quenines et leur représentation spirale, Mathématiques et Sciences Humaines, Centre de Mathématique Sociale et de statistique, EPHE, 2008, 184 (4), pp. 9-23, hal-00188240.
G. Esposito-Farese, C program
Index entries for sequences related to the Josephus Problem

Not to be confused with Queneau's "s-additive sequences", see A003044.
A005384 is a subsequence.
Union of A163782 (Josephus_2-primes) and A163781 (dual Josephus_2-primes); also the union of A163777 (Archimedes_0-primes) and A163778 (Archimedes_1-primes); also the union of A071642/2 (shuffle primes) and A163776/2 (dual shuffle primes). - Peter R. J. Asveld, Aug 17 2009
Cf. A216371, A003558 (for which a(n) == n).

QD:= proc(n) local i;
  if n::even then map(op,[seq([n-i,i+1],i=0..n/2-1)])
  else map(op, [seq([n-i,i+1],i=0..(n-1)/2-1),[(n+1)/2]])
  fi
end proc:
select(n -> GroupTheory:-PermOrder(Perm(QD(n)))=n, [$1..1000]); # Robert Israel, May 01 2016

a[p_] := Sum[Cos[2^n Pi/((2 p + 1) )], {n, 1, p}];
Select[Range[500],Reduce[a[#] == -1/2, Rationals] &] (* Gerry Martens, May 01 2016 *)

is(n)=
{
    if (n==1, return(1));
    my( m=n%4 );
    if ( m==4, return(0) );
    my(p=2*n+1, r=znorder(Mod(2,p)));
    if ( !isprime(p), return(0) );
    if ( m==3 && r==n, return(1) );
    if ( r==2*n, return(1) ); \\ r == 1 or 2
    return(0);
}
for(n=1,10^3, if(is(n),print1(n,", ")) );
\\ Joerg Arndt, May 02 2016

a(n) = (A216371(n)-1)/2. - L. Edson Jeffery, Dec 18 2012

a(n) >> n log n, and on the Bateman-Horn-Stemmler conjecture a(n) << n log^2 n. I imagine a(n) ≍ n log n, and numerics suggest that perhaps a(n) ~ kn log n for some constant k (which seems to be around 1.122). - Charles R Greathouse IV, Aug 02 2023

A233547 a(n) = |{0 < k < n/2: phi(k)phi(n-k) - 1 and phi(k)phi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 1, 3, 4, 3, 2, 3, 2, 3, 1, 1, 2, 1, 5, 2, 3, 1, 2, 1, 1, 3, 4, 5, 4, 3, 2, 3, 2, 5, 2, 5, 5, 3, 5, 3, 1, 5, 3, 7, 6, 3, 2, 4, 7, 5, 1, 4, 6, 6, 5, 2, 4, 6, 9, 9, 6, 8, 5, 8, 8, 6, 6, 9, 4, 8, 6, 8, 5, 7, 9, 7, 9, 5, 7, 3, 9, 5, 6, 7, 7, 10, 5, 12, 7, 5, 7, 5, 7, 5, 7, 8, 4, 7, 13

Offset: 1

Zhi-Wei Sun, Dec 12 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 5.
(ii) For any n > 3, sigma(k)*phi(n-k) - 1 and sigma(k)*phi(n-k) + 1 are both prime for some 0 < k < n, where sigma(k) is the sum of all (positive) divisors of k.
(iii) For any n > 5 not equal to 35, there is a positive integer k < n such that phi(k)*phi(n-k) - 1 is a Sophie Germain prime.
Note that part (i) implies the twin prime conjecture. We have verified it for n up to 10^7.

			a(6) = 1 since phi(1)*phi(5) = 1*4 = 4 with 4 - 1 and 4 + 1 twin primes.
a(8) = 1 since phi(1)*phi(7) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(16) = 1 since phi(2)*phi(14) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(17) = 1 since phi(3)*phi(14) = 2*6 = 12 with 12 - 1 and 12 + 1 twin primes.
a(19) = 1 since phi(1)*phi(18) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(23) = 1 since phi(2)*phi(21) = 1*12 = 12 with 12 - 1 and 12 + 1 twin primes.
a(25) = 1 since phi(11)*phi(14) = 10*6 = 60 with 60 - 1 and 60 + 1 twin primes.
a(26) = 1 since phi(7)*phi(19) = 6*18 = 108 with 108 - 1 and 108 + 1 twin primes.
a(42) = 1 since phi(14)*phi(28) = 6*12 = 72 with 72 - 1 and 72 +1 twin primes.
a(52) = 1 since phi(14)*phi(38) = 6*18 = 108 with 108 - 1 and 108 + 1 twin primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A001359, A005384, A006512, A014574, A232270, A233544.

TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
a[n_]:=Sum[If[TQ[EulerPhi[k]*EulerPhi[n-k]],1,0],{k,1,(n-1)/2}]
Table[a[n],{n,1,100}]

A035096 a(n) is the smallest k such that prime(n)*k+1 is prime.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 6, 10, 2, 2, 10, 4, 2, 4, 6, 2, 12, 6, 4, 8, 4, 4, 2, 2, 4, 6, 6, 6, 10, 2, 4, 2, 6, 4, 8, 6, 10, 4, 14, 2, 2, 6, 2, 4, 18, 4, 10, 12, 24, 12, 2, 2, 6, 2, 6, 6, 8, 6, 4, 2, 6, 2, 4, 6, 6, 26, 6, 10, 6, 10, 14, 2, 6, 4, 12, 12, 24, 6, 8, 4, 2, 10, 2, 4, 10, 2, 8, 30

Offset: 1

Labos Elemer

nonn

These arithmetic progressions have prime differences. Note that both the terms of generated by this k values and the differences are primes as well.
This is one possible generalization of "the least prime problem in special arithmetic progressions" when n in the nk+1 form is replaced by n-th prime number.
Note that Dirichlet's theorem on primes in arithmetic progressions implies that a(n) always exists. - Max Alekseyev, Jul 11 2008
If a(n)=2, prime(n) is a Sophie Germain prime (A005384). Among the first 10^6 terms, the largest is a(330408) = 234. - Zak Seidov, Jan 28 2012

			a(15)=6 because the 15th prime is 47, and the smallest k such that 47k+1 is prime is k=6, for which 47k+1=283.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Math, Dirichlet's theorem.
Index entries for sequences related to primes in arithmetic progressions.

Smallest k such that k*n+1 is prime is A034693.
Sophie Germain primes are in A005384.
Cf. A000040, A035095. - Zak Seidov, Dec 27 2013
Cf. A117673.

S:=[];
k:=1;
for n in [1..90] do
  while not IsPrime(k*NthPrime(n)+1) do
       k:=k+1;
  end while;
  Append(~S, k);
  k:=1;
end for;
S; // Bruno Berselli, Apr 18 2013

Reap[Sow[1]; Do[p = Prime[n]; k = 2; While[! PrimeQ[k*p + 1], k = k + 2]; Sow[k], {n, 2, 10^4}]][[2, 1]] (* Zak Seidov, Jan 28 2012 *)
f[n_] := Block[{p = Prime@ n}, q = 1 + 2p; While[ !PrimeQ@ q, q += 2p]; (q - 1)/p]; f[1] = 1; Array[f, 88] (* Robert G. Wilson v, Dec 27 2014 *)

a(n) = if(n == 1, 1, my(t = 2*prime(n), m = t + 1); while(!isprime(m), m += t); 2*(m - 1)/t); \\ Amiram Eldar, Mar 19 2025

a(n) = (A035095(n)-1)/A000040(n). - Zak Seidov, Dec 27 2013

A059788 a(n) = largest prime < 2*prime(n).

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 37, 43, 53, 61, 73, 79, 83, 89, 103, 113, 113, 131, 139, 139, 157, 163, 173, 193, 199, 199, 211, 211, 223, 251, 257, 271, 277, 293, 293, 313, 317, 331, 337, 353, 359, 379, 383, 389, 397, 421, 443, 449, 457, 463, 467, 479, 499, 509, 523

Offset: 1

Labos Elemer, Feb 22 2001

nonn

Also, smallest member of the first pair of consecutive primes such that between them is a composite number divisible by the n-th prime. - Amarnath Murthy, Sep 25 2002

Except for its initial term, A006992 is a subsequence based on iteration of n -> A151799(2n). The range of this sequence is a subset of A065091. - M. F. Hasler, May 08 2016

			n=18: p(18)=61, so a(18) is the largest prime below 2*61=122, which is 113.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A005382, A005383, A005384, A005385, A055496, A006992, A052248.

with(numtheory):
A059788 := proc(n)
    prevprime(2*ithprime(n)) ;
end proc:
seq(A059788(n),n=1..50) ; # R. J. Mathar, May 08 2016

a[n_] := Prime[PrimePi[2Prime[n]]]
NextPrime[2*Prime[Range[100]], -1] (* Zak Seidov, May 08 2016 *)

a(n) = precprime(2*prime(n)); \\ Michel Marcus, May 08 2016

a(n) = A007917(A100484(n)). - R. J. Mathar, May 08 2016

A156592 Product pq of two primes with q = 2p + 1.

Original entry on oeis.org

10, 21, 55, 253, 1081, 1711, 3403, 5671, 13861, 15931, 25651, 34453, 60031, 64261, 73153, 108811, 114481, 126253, 158203, 171991, 258121, 351541, 371953, 392941, 482653, 518671, 703891, 822403, 853471, 869221, 933661, 1034641, 1104841

Offset: 1

Reinhard Zumkeller, Feb 10 2009

nonn

Subsequence of A068443.

Products of Sophie Germain primes p with their corresponding safe primes 2p+1. The smallest prime factor of a(n) is (a(n) - phi(a(n)))/3 and the largest prime factor of a(n) is 2(a(n) - phi(a(n)))/3 + 1. - Wesley Ivan Hurt, Oct 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..4000
Math Marathon (MM95) (in Russian) [from Vladimir Joseph Stephan Orlovsky, May 12 2010]

Cf. A005384, A005385. Subset of A001358.

lst={};Do[p=Prime[n];q=2*p+1;If[PrimeQ[q],AppendTo[lst,p*q]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)

a(n) = A005384(n)*A005385(n).

cp's OEIS Frontend

A111206 Semi-Sophie Germain semiprimes: semiprimes which are the product of Sophie Germain primes.

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A124485 Numbers n such that 2n-1 and 4n-1 are primes.

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A209253 Number of ways to write 2n-1 as the sum of a Sophie Germain prime and a practical number.

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A002174 Values taken by reduced totient function psi(n).

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A059761 Initial primes of Cunningham chains of first type with length exactly 2. Primes in A059453 that survive as primes only one "2p-1 iteration", forming chains of exactly 2 terms.

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A054639 Queneau numbers: numbers n such that the Queneau-Daniel permutation {1, 2, 3, ..., n} -> {n, 1, n-1, 2, n-2, 3, ...} is of order n.

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A233547 a(n) = |{0 < k < n/2: phi(k)*phi(n-k) - 1 and phi(k)*phi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.

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A233547 a(n) = |{0 < k < n/2: phi(k)phi(n-k) - 1 and phi(k)phi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.

A156592 Product pq of two primes with q = 2p + 1.