A002515 -id:A002515 - cp's OEIS frontend

A101796 Primes of the form 8k-1 such that 4k-1, 16k-1 and 32k-1 are also primes.

359, 719, 5399, 7079, 24239, 34319, 54959, 107279, 115679, 126839, 142799, 149399, 164999, 175079, 202799, 214559, 215399, 225839, 244199, 245639, 253679, 254279, 266999, 278879, 333479, 335519, 459479, 507359, 508559

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 359 is a term.

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

Cf. A002515, A101794, A101795, A101797, A101798.
Subsequence of A007522 and A101792.
Subsequence: A101996.

8 * Select[Range[10^5], And @@ PrimeQ[2^Range[2, 5]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)

is(k) = if(k % 8 == 7, my(m = k\8 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 8*A101794(n) - 1 = 2*A101795(n) + 1. - Amiram Eldar, May 13 2024

A101797 Primes of the form 16k-1 such that 4k-1, 8k-1 and 32k-1 are also primes.

Original entry on oeis.org

719, 1439, 10799, 14159, 48479, 68639, 109919, 214559, 231359, 253679, 285599, 298799, 329999, 350159, 405599, 429119, 430799, 451679, 488399, 491279, 507359, 508559, 533999, 557759, 666959, 671039, 918959, 1014719, 1017119, 1148879

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 719 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002515, A101794, A101795, A101796, A101798.
Subsequence of A127576 and A101793.
Subsequence: A101997.

16#-1&/@Select[Range[80000],AllTrue[#*2^Range[2,5]-1,PrimeQ]&] (* Harvey P. Dale, Apr 25 2015 *)

is(k) = if(k % 16 == 15, my(m = k\16 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 16*A101794(n) - 1 = 4*A101795(n) + 3 = 2*A101796(n) + 1. - Amiram Eldar, May 13 2024

A188130 Primes p such that 6p+1 divides the Mersenne number M(p)=A000225(p).

Original entry on oeis.org

5, 37, 73, 233, 397, 461, 557, 577, 601, 761, 1013, 1321, 1361, 1381, 1453, 1693, 1777, 1993, 2417, 2593, 2621, 2897, 3037, 3181, 3457, 3581, 3593, 4001, 4273, 4441, 4517, 4597, 4801, 4813, 4861, 4933, 5197, 5393, 5557, 5717, 5801, 6173, 6277, 6353, 6373, 6841, 6977, 7573, 7853, 7901, 8353, 8377, 9613, 10321, 10357

Offset: 1

M. F. Hasler, Mar 21 2011

nonn

These primes are such that p=1 (mod 4) and 6p+1 is prime, but there are other primes with these properties (13, 17, ...) not in this sequence.

There are no primes p such that 4p+1 divides M(p), but those for which 2p+1 divides M(p) are the Lucasian primes A002515, and those for which 10p+1 divides M(p) are listed in A188133.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1313 from M. F. Hasler)

Cf. A002515, A188133.

Primes in A038844.

Select[Range[10^4], PrimeQ[#] && PowerMod[2, #, 6# + 1] == 1 &] (* Amiram Eldar, Nov 13 2019 *)

forprime(p=1,1e5,Mod(2,p*6+1)^p-1||print1(p", "))

A101791 Primes of the form 4k-1 such that 8k-1 and 16*k-1 are also primes.

Original entry on oeis.org

11, 179, 359, 719, 1019, 1031, 1451, 1511, 1811, 1931, 2459, 2699, 2819, 3491, 3539, 3911, 5399, 6131, 7079, 7151, 10691, 11171, 11471, 12119, 12899, 12959, 16811, 17159, 18191, 19319, 19991, 20411, 21011, 21179, 22271, 23099, 23819

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*3-1 = 11, 8*3-1 = 23 and 16*3-1 = 47 are primes, so 11 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002515, A101790, A101792, A101793.
Subsequence of A002145.
Subsequences: A101795, A101995.

p4816Q[n_]:=Module[{nn=(n+1)/4},And@@PrimeQ[{n,8nn-1,16nn-1}]]; Select[ 4*Range[6000]-1,p4816Q] (* Harvey P. Dale, Nov 25 2011 *)

is(k) = if(k % 4 == 3, my(m = k\4 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 4*A101790(n) - 1. - Amiram Eldar, May 13 2024

A101792 Primes of the form 8k-1 such that 4k-1 and 16*k-1 are also primes.

Original entry on oeis.org

23, 359, 719, 1439, 2039, 2063, 2903, 3023, 3623, 3863, 4919, 5399, 5639, 6983, 7079, 7823, 10799, 12263, 14159, 14303, 21383, 22343, 22943, 24239, 25799, 25919, 33623, 34319, 36383, 38639, 39983, 40823, 42023, 42359, 44543, 46199, 47639, 48479, 49103, 54959

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*3 - 1 = 11, 8*3 - 1 = 23 and 16*3 - 1 = 47 are primes, so 23 is a term.

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

Cf. A002515, A101790, A101791, A101793.
Subsequence of A007522.
Subsequences: A101796, A101996.

8 * Select[Range[10^4], And @@ PrimeQ[2^Range[2, 4]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)

for(k=1,7000,if(isprime(8*k-1)&&isprime(4*k-1)&&isprime(16*k-1),print1(8*k-1,", "))) \\ Hugo Pfoertner, Sep 07 2021

a(n) = 8*A101790(n) - 1 = 2*A101791(n) + 1. - Amiram Eldar, May 13 2024

A101798 Primes of the form 32k-1 such that 4k-1, 8k-1 and 16k-1 are also primes.

Original entry on oeis.org

1439, 2879, 21599, 28319, 96959, 137279, 219839, 429119, 462719, 507359, 571199, 597599, 659999, 700319, 811199, 858239, 861599, 903359, 976799, 982559, 1014719, 1017119, 1067999, 1115519, 1333919, 1342079, 1837919, 2029439, 2034239

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 1439 is a term.

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

Cf. A002515, A101794, A101795, A101796, A101797.
Subsequence of A127578.
Subsequence: A101998.

32 * Select[Range[10^5], And @@ PrimeQ[2^Range[2, 5]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)
Select[Prime[Range[200000]],Mod[#,32]==31&&AllTrue[{4,8,16} (#+1)/32-1,PrimeQ]&] (* Harvey P. Dale, Feb 20 2025 *)

is(k) = if(k % 32 == 31, my(m = k\32 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 32*A101794(n) - 1 = 8*A101795(n) + 7 = 4*A101796(n) + 3 = 2*A101797(n) + 1. - Amiram Eldar, May 13 2024

A103579 Sophie Germain primes that are not Lucasian primes: primes p not 3 (mod 4) such that 2p + 1 is prime.

Original entry on oeis.org

2, 5, 29, 41, 53, 89, 113, 173, 233, 281, 293, 509, 593, 641, 653, 761, 809, 953, 1013, 1049, 1229, 1289, 1409, 1481, 1601, 1733, 1889, 1901, 1973, 2069, 2129, 2141, 2273, 2393, 2549, 2693, 2741, 2753, 2969, 3329, 3389, 3413, 3449, 3593, 3761, 3821, 4073, 4349, 4373, 4409, 4481, 4733, 4793, 5081

Offset: 1

Jonathan Vos Post, Mar 23 2005

For n > 1, the prime 2*a(n) + 1 is the smallest prime divisor of (2^a(n) + 1)/3. - Emmanuel Vantieghem, Aug 12 2018
Primes p such that 2*p+1 divides 2^p+1. - Hilko Koning, Sep 21 2021
Subset of Josephus_2 primes {A163782} that are themselves also prime. - Joe Nellis, Dec 27 2022

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Luis Henri Gallardo, Bell Numbers Modulo p, Appl. Math. E-Notes (2023) Vol. 23, 40-48. See p. 43.

Cf. A002144, A002515, A005384, A163782.

select(t -> isprime(t) and isprime(2*t+1),[2,seq(4*k+1,k=1..10000)]); # Robert Israel, May 20 2015

Select[Prime[Range[500]], PrimeQ[2#+ 1 ] && Mod[#, 4] != 3 &] (* Harvey P. Dale, Jun 15 2013 *)
Select[4Range[100] + 1, PrimeQ[#] && PrimeQ[2# + 1] &] (* Alonso del Arte, Jun 01 2019 *)

forprime(p=2,10^4,if((p%4!=3)&&isprime(2*p+1),print1(p,", "))); \\ Joerg Arndt, Nov 18 2014

More terms from Vladimir Joseph Stephan Orlovsky, Jul 07 2009

A158036 Integer solutions f for f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) with n an integer.

Original entry on oeis.org

3, 8287, 32547981403, 3374074914839397834392750148706282243018046503, 107547872626305931371847778721098686654377801057464206176785452350259573207, 4568366860875634575966528292411682488942909674818941246717098803707597353756388768388059303363024343431

Offset: 1

Reikku Kulon, Mar 11 2009

nonn

8287 = 129 * 64 + 31 = 257 * 32 + 63 is prime. A158034 (values of n) is often prime. A158035 (2n + 1) appears to be always prime.

See A235540 for nonprimes in A158034. - Reinhard Zumkeller, Nov 17 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..32

Cf. A158034, A158035 (n, 2n + 1)
Cf. A002515 (Lucasian primes)
Cf. A145918 (exponential Sophie Germain primes)
Cf. A235540.

a158036 = (\x -> (4^x - 2^x + 8*x^2 - 2) `div` (2*x*(2*x + 1))) . a158034
-- Reinhard Zumkeller, Nov 17 2014

A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.

Original entry on oeis.org

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, 1541, 1559

Offset: 1

Reikku Kulon, Jun 21 2009

Near superset of the Sophie Germain primes (A005384), excluding 2 and 3: 2n + 1 is prime. Nearly all members of this sequence are also prime, but four members less than 10000 are composite: 1541 = 23 * 67, 2465 = 5 * 17 * 29, 3281 = 17 * 193, and 4961 = 11^2 * 41.
The congruence of n modulo 4 is evenly distributed between 1 and 3. n is congruent to 5 (mod 6) for all n less than two billion.
This sequence has roughly twice the density of the sequence (A158034) corresponding to the Diophantine equation
f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)),
and contains most members of that sequence. Those it does not contain are composite and often congruent to 3 (mod 6).
Composite terms appear to predominantly belong to A262051. - Bill McEachen, Aug 29 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A161897 A000040, A002515, A005384, A158034, A158035, A158036, A145918, A002943.

a161896 n = a161896_list !! (n-1)
a161896_list = [x | x <- [1..],
                    (9^x - 3*3^x - 4*x) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

is(n)=my(m=2*n*(2*n+1),t=Mod(3,m)^n); t^2-3*t==4*n \\ Charles R Greathouse IV, Nov 25 2014

A186283 Least number k such that k*n+1 is a prime dividing 2^n-1.

Original entry on oeis.org

1, 2, 1, 6, 1, 18, 2, 8, 1, 2, 1, 630, 3, 2, 1, 7710, 1, 27594, 2, 6, 1, 2, 10, 24, 105, 9728, 1, 8, 1, 69273666, 8, 18166, 1285, 2, 1, 6, 4599, 2, 1, 326, 1, 10, 2, 14, 1, 50, 2, 90462791808, 5, 2, 1, 120, 1615, 16, 2, 568, 1, 3050, 1, 37800705069076950, 11545611, 2, 4, 126, 1, 2891160, 2, 145690999102, 1

Offset: 2

Bill McEachen, Feb 16 2011

nonn

The smallest prime factor of 2^n-1 of the form k*n+1 is A186522(n).

By Fermat's little theorem, a(n) = 1 if and only if n+1 is an odd prime. Further, for prime p, a(p) = 2 if and only if p is in A002515. - Thomas Ordowski, Sep 03 2017

			For n=8, 2^n-1 = 255 = 3 * 5 * 17.  The smallest prime factor of the form k*n+1 is 17 = 2*8+1. Hence, a(8) = 2.

Kenneth H. Rosen, Elementary Number Theory and Its Applications, 3rd Ed, Theorem 6.12, p. 225

Max Alekseyev, Table of n, a(n) for n = 2..1236 (terms to a(300) from David A. Corneth)
Will Edgington, Mersenne Page [from Internet Archive Wayback Machine]
Eric W. Weisstein, MathWorld: Mersenne Number
Eric W. Weisstein, MathWorld: Mersenne Prime

Cf. A000225, A049479, A079324.

Table[p=First/@FactorInteger[2^n-1]; (Select[p, Mod[#1,n] == 1 &, 1][[1]] - 1)/n, {n, 2, 70}]

a(n) = {if(isprime(n+1),return(1)); my(f = factor(2^n - 1)[,1]); for(i=1,#f, if(f[i]%n == 1, return((f[i]-1) / n)))} \\ David A. Corneth, Sep 03 2017

a(n) = (A186522(n)-1)/n.

cp's OEIS Frontend

A101796 Primes of the form 8*k-1 such that 4*k-1, 16*k-1 and 32*k-1 are also primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A101797 Primes of the form 16*k-1 such that 4*k-1, 8*k-1 and 32*k-1 are also primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A188130 Primes p such that 6p+1 divides the Mersenne number M(p)=A000225(p).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A101791 Primes of the form 4*k-1 such that 8*k-1 and 16*k-1 are also primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A101792 Primes of the form 8*k-1 such that 4*k-1 and 16*k-1 are also primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A101798 Primes of the form 32*k-1 such that 4*k-1, 8*k-1 and 16*k-1 are also primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A103579 Sophie Germain primes that are not Lucasian primes: primes p not 3 (mod 4) such that 2p + 1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A158036 Integer solutions f for f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) with n an integer.

Views

A101796 Primes of the form 8k-1 such that 4k-1, 16k-1 and 32k-1 are also primes.

A101797 Primes of the form 16k-1 such that 4k-1, 8k-1 and 32k-1 are also primes.

A101791 Primes of the form 4k-1 such that 8k-1 and 16*k-1 are also primes.

A101792 Primes of the form 8k-1 such that 4k-1 and 16*k-1 are also primes.

A101798 Primes of the form 32k-1 such that 4k-1, 8k-1 and 16k-1 are also primes.