A053176 -id:A053176 - cp's OEIS frontend

A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

2, 3, 29, 41, 53, 89, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Except for 2 and 3 these primes are congruent to 5 or 11 modulo 12.

Introducing terms of Cunningham chains of first kind.

			89 is a term because (89-1)/2 = 44 is not prime, but 2*89 + 1 = 179 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A059456.

Cf. A005385, A053176, A059452, A059453, A059454, A059455, A007700, A005602, A023272, A023302, A023330.

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 *)
Select[Prime[Range[300]],PrimeQ[2#+1]&&!PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Nov 10 2017 *)

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

from itertools import count, islice
from sympy import isprime, prime
def A059453_gen(): # generator of terms
    return filter(lambda p:not isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059453_list = list(islice(A059453_gen(),10)) # Chai Wah Wu, Jul 12 2022

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

A059766 Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.

Original entry on oeis.org

89, 63419, 127139, 405269, 810809, 1069199, 1178609, 1333889, 1598699, 1806089, 1958249, 2606069, 2848949, 3241289, 3339989, 3784199, 3962039, 4088879, 4444829, 4664249, 4894889, 4897709, 5132999, 5215499, 5238179, 6026309, 6059519, 6088529, 6490769, 6676259

Offset: 1

Labos Elemer, Feb 21 2001

nonn

Special terms of A059453. Not identical to A023330 of which 1122659, 2164229, 2329469, ..., etc. are omitted since they have exact length 7 or larger.

Unsafe primes starting complete chains of length 6.

			89 is a term because (89-1)/2 = 44 and 64*89+63 = 5759 = 13*443 are composites, while 89, 179, 359, 719, 1439, and 2879 are primes.
1122659 is not a term because it initiates a chain of length 7.
4658939 is not a term because (4658939-1)/2 = 2329469 is prime. - _Sean A. Irvine_, Oct 09 2022

Amiram Eldar, Table of n, a(n) for n = 1..1000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Cf. A005602, A053176, A059500, A057331, A059688, A059690.

Entry revised by N. J. A. Sloane Apr 01 2006

a(12) onward corrected and extended by Sean A. Irvine, Oct 09 2022

A075712 Rearrangement of primes into Germain groups (or Cunningham chains).

Original entry on oeis.org

2, 5, 11, 23, 47, 3, 7, 13, 17, 19, 29, 59, 31, 37, 41, 83, 167, 43, 53, 107, 61, 67, 71, 73, 79, 89, 179, 359, 719, 1439, 2879, 97, 101, 103, 109, 113, 227, 127, 131, 263, 137, 139, 149, 151, 157, 163, 173, 347, 181, 191, 383, 193, 197, 199, 211, 223, 229, 233

Offset: 1

Zak Seidov, Oct 03 2002

In each group, p(i+1) = 2*p(i)+1.

The groups are also known as Cunningham chains of the first kind.

			The groups are:
{2, 5, 11, 23, 47},
{3, 7},
{13},
{17},
{19},
{29, 59},
{31},
{37},
{41, 83, 167},
{43},
{53, 107},
{61},
{67},
{71},
{73},
{79},
{89, 179, 359, 719, 1439, 2879},
{97},
{101},
{103},
{109},
{113, 227},
{127},
{131, 263},
{137},
{139},
...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005384, A059452, A059453, A059455, A059456, A053176.
See also A181697.
See A059456 for initial terms, A338945 for lengths.

Block[{a = {2}, j = 1, k, p}, Do[k = j; If[PrimeQ@ a[[-1]], AppendTo[a, 2 a[[-1]] + 1], While[! FreeQ[a, Set[p, Prime[k]]], k++]; j++; Set[a, Append[a[[1 ;; -2]], p]]], 10^3]; a] (* Michael De Vlieger, Nov 17 2020 *)

first(n) = my(res=List([2,5,11,23,47])); forprime(p=3, oo, if(!isprime((p-1)>>1), listput(res,p); c = 2*p+1; while(isprime(c), listput(res,c); c=2*c+1)); if(#res>n,return(res))); res \\ David A. Corneth, Nov 13 2021

Edited by N. J. A. Sloane, Nov 13 2021

More terms from David A. Corneth, Nov 13 2021

A156542 Number of distinct Sophie Germain prime factors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 0, 1, 2, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 2, 2, 2, 0, 2, 0, 2, 1, 1, 1, 2, 2, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 0, 2, 1, 1, 1, 2, 0, 1, 2, 2, 0, 2, 0, 1, 2

Offset: 1

Reinhard Zumkeller, Feb 10 2009

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000720, A001221, A005384, A053176, A156541, A156543, A101264.

[0] cat [&+[#PrimesInInterval(2*p,2*p+1):p in PrimeDivisors(n)]:n in [2..100]]; // Marius A. Burtea, Aug 25 2019

with(numtheory): A101264:=p-> pi(2*p+1)-pi(2*p): seq(add(A101264(p), p in factorset(n)), n=1..100); # Ridouane Oudra, Aug 25 2019

Join[{0},Table[Count[FactorInteger[n][[All,1]],?(PrimeQ[2#+1]&)],{n,2,110}]] (* _Harvey P. Dale, Apr 05 2020 *)

a(n) <= A001221(n).
a(A156541(n)) = A001221(A156541(n)); a(A156543(n)) = 0.
a(A005384(n)) = 1; a(A053176(n)) = 0.
a(n) = Sum_{p|n} A101264(p), where p is a prime. - Ridouane Oudra, Aug 25 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A005384(k) (see A005384 for an estimate of this sum). - Amiram Eldar, Jun 03 2024

A068497 Primes p such that 2p+1 and 2p-1 are composites.

Original entry on oeis.org

13, 17, 43, 47, 59, 61, 67, 71, 73, 101, 103, 107, 109, 127, 137, 149, 151, 163, 167, 181, 193, 197, 223, 227, 241, 257, 263, 269, 277, 283, 311, 313, 317, 347, 349, 353, 373, 383, 389, 397, 401, 409, 421, 433, 449, 457, 461, 463, 467, 479, 487, 503, 521

Offset: 1

Benoit Cloitre, Mar 25 2002

Subsequence of A053176. - Michel Marcus, Jan 12 2015

The sequence is infinite. Among others it contains all the primes of the form 15m+/-2. - Emmanuel Vantieghem, Sep 19 2016

G. C. Greubel, Table of n, a(n) for n = 1..10000

Filtered([1..530],p->IsPrime(p) and not IsPrime(2*p+1) and not IsPrime(2*p-1)); # Muniru A Asiru, Oct 16 2018

[p: p in PrimesUpTo(600) | not IsPrime(2*p+1) and not IsPrime(2*p-1)]; // Vincenzo Librandi, Jan 20 2015

select(p->isprime(p) and not isprime(2*p+1) and not isprime(2*p-1),[$1..530]); # Muniru A Asiru, Oct 16 2018

lst={};Do[p=Prime[n];If[ !PrimeQ[2*p-1]&&!PrimeQ[2*p+1],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
Select[Prime[Range[500]], ! PrimeQ[2*# - 1] && ! PrimeQ[2*# + 1] &] (* G. C. Greubel, Oct 15 2018 *)

listp(nn) = {forprime(p=2, nn, if (!isprime(2*p-1) && !isprime(2*p+1), print1(p, ", ")););} \\ Michel Marcus, Jan 12 2015

A043297 Primes p such that B(4*p) has denominator 30 where B(2n) are the Bernoulli numbers.

Original entry on oeis.org

2, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 181, 197, 211, 223, 227, 229, 241, 257, 263, 269, 271, 283, 311, 313, 317, 331, 337, 347, 349, 353, 367, 379, 383, 389, 397, 401, 421, 439, 449, 457, 461, 463, 467, 479, 503, 521

Offset: 1

Benoit Cloitre, Mar 24 2002

Complement of A087634, primes p such that phi(k) = 4p has a solution, where phi is Euler's totient function.
The sequences a(n), A005384 and A023212 form a partition of the set of primes > 3: Using von Staudt-Clausen formula the divisors of 4p increased by 1 are {2,3,5,p+1,2p+1,4p+1}, p+1 is clearly an even number, and if 2p+1 and 4p+1 are not prime, then denom(B(4p))=30. - Enrique Pérez Herrero, Aug 15 2011
Also 2 with the primes p such that both 2*p+1 and 4*p+1 are composite: A210684. For these numbers k > 2 the equation: phi(n)=k*tau(n), where phi is A000010 and tau is A000005, has no solutions: A112954(a(n))=0. - Enrique Pérez Herrero, May 12 2012

E. Pérez Herrero, Table of n, a(n) for n=1..30000
Wikipedia, Von Staudt-Clausen theorem

Cf. A051225, A053176, A087634.
Cf. A005384, A023212.
Cf. A210684.

Select[Prime[Range[100]], Denominator[BernoulliB[4# ]]==30&] (* T. D. Noe, Feb 19 2004 *)
Select[Prime[Range[100]],!PrimeQ[4#+1]&&!PrimeQ[2#+1]||(#==2)&] (* Enrique Pérez Herrero, Aug 16 2011 *)

A307390 Primes p such that 2*p-1 is not prime.

Original entry on oeis.org

5, 11, 13, 17, 23, 29, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 223, 227, 233, 239, 241, 251, 257, 263, 269, 277, 281, 283, 293, 311, 313, 317, 347, 349, 353, 359, 373, 383, 389, 397, 401, 409, 419, 421

Offset: 1

Robert Israel, Apr 17 2019

nonn

Primes not in A005382.

			a(3) = 13 is in the sequence because 13 is prime but 2*13-1 = 25 is not.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005382, A053176, A109274.

Includes A007528.

select(t -> isprime(t) and not isprime(2*t-1), [seq(i,i=3..1000,2)]);

a(n) = A109274(n) + 1. - Bhavik Mehta, Aug 14 2024

A053177 Odd composite k such that (k-1)/2 is prime.

Original entry on oeis.org

15, 27, 35, 39, 63, 75, 87, 95, 119, 123, 135, 143, 147, 159, 195, 203, 207, 215, 219, 255, 275, 279, 299, 303, 315, 327, 335, 363, 387, 395, 399, 423, 447, 455, 459, 483, 515, 527, 539, 543, 555, 567, 615, 623, 627, 635, 663, 675, 695, 699, 707, 735, 747

Offset: 1

Enoch Haga, Feb 29 2000

Composite numbers produced in A053176.

			a(3)=35 and 35-1=34, 34/2=17, prime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Cf. A053176, A005385, A005382, A259978.

Select[2 Prime@ Range@ 74 + 1, CompositeQ] (* Michael De Vlieger, Jul 13 2015 *)
Select[Range[1,801,2],CompositeQ[#]&&PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Apr 01 2019 *)

main(size)={my(v=vector(size),i,t=1);for(i=1, size, while(isprime(2*prime(t)+1), t++); v[i]=2*prime(t)+1;t++;);return(v)} /* Anders Hellström, Jul 13 2015 */

From the composite, subtract 1, divide by 2 and result is a prime.

Definition clarified by Peter Munn, Oct 26 2017

A230117 Primes p such that 2p+1 is prime and 2p+3 is not prime.

Original entry on oeis.org

3, 11, 23, 41, 83, 131, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 593, 641, 653, 683, 719, 761, 911, 953, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1511, 1601, 1811, 1889, 1901, 1931, 1973, 2003, 2039, 2069, 2141

Offset: 1

Vincenzo Librandi, Oct 10 2013

Intersection of A005384 and A163769. - Felix Fröhlich, Jan 14 2017

			23 is in the sequence because 2*23+1=47 (prime) and 2*23+3=49 (not prime).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005384, A023204, A053176, A126107, A163769, A230039.

[p: p in PrimesUpTo(2500)| IsPrime(2*p+1) and not IsPrime(2*p+3)];

Select[Range[10^6],PrimeQ[#]&& PrimeQ[2#+1]&&!PrimeQ[2#+3]&]

is(n) = ispseudoprime(n) && ispseudoprime(2*n+1) && !ispseudoprime(2*n+3) \\ Felix Fröhlich, Jan 14 2017

A230039 Primes p such that 2p+1 is not prime and 2p+3 is prime.

Original entry on oeis.org

7, 13, 17, 19, 43, 47, 67, 73, 97, 127, 137, 139, 157, 167, 193, 197, 199, 223, 227, 229, 269, 277, 283, 307, 337, 349, 353, 379, 383, 397, 409, 439, 463, 467, 487, 503, 523, 547, 557, 563, 599, 607, 613, 617, 643, 647, 739, 773, 797, 827, 853, 859, 887, 929

Offset: 1

Vincenzo Librandi, Oct 10 2013

Intersection of A023204 and A053176. - Felix Fröhlich, Jan 14 2017

			43 is in the sequence because 2*43+1=87 (not prime) and 2*43+3=89 (prime).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005384, A023204, A053176, A126107, A163769, A230117.

[p: p in PrimesUpTo(2500)|not IsPrime(2*p+1) and IsPrime(2*p+3)];

Select[Range[10^5],PrimeQ[#]&& !PrimeQ[2#+1]&& PrimeQ[2#+3]&]

is(n) = ispseudoprime(n) && !ispseudoprime(2*n+1) && ispseudoprime(2*n+3) \\ Felix Fröhlich, Jan 14 2017

cp's OEIS Frontend

A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A059766 Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A075712 Rearrangement of primes into Germain groups (or Cunningham chains).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A156542 Number of distinct Sophie Germain prime factors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A068497 Primes p such that 2*p+1 and 2*p-1 are composites.

Views

Author

Keywords

Comments

Links

Programs

GAP

Magma

Maple

Mathematica

PARI

A043297 Primes p such that B(4*p) has denominator 30 where B(2n) are the Bernoulli numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A307390 Primes p such that 2*p-1 is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A053177 Odd composite k such that (k-1)/2 is prime.

Views

A068497 Primes p such that 2p+1 and 2p-1 are composites.

A230117 Primes p such that 2p+1 is prime and 2p+3 is not prime.

A230039 Primes p such that 2p+1 is not prime and 2p+3 is prime.