A005382 -id:A005382 - cp's OEIS frontend

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

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

A005939 Pseudoprimes to base 10.

Original entry on oeis.org

9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187, 4521, 5461, 6533, 6541, 6601, 7107, 7471, 7777, 8149, 8401, 8911, 10001, 11111, 11169, 11649, 12403, 12801, 13833, 13981, 14701, 14817, 14911, 15211

Offset: 1

N. J. A. Sloane

nonn

This sequence is a subsequence of A121014 & A121912. In fact the terms are composite terms n of these sequences such that gcd(n,10)=1. Theorem: If both numbers q & 2q-1 are primes(q is in the sequence A005382) and n=q*(2q-1) then 10^(n-1) == 1 (mod n) (n is in the sequence A005939) iff mod(q, 20) is in the set {1, 7, 19}. 91,703,12403,38503,79003,188191,269011,... are such terms. - Farideh Firoozbakht, Sep 15 2006
Composite numbers n such that 10^(n-1) == 1 (mod n). - Michel Lagneau, Feb 18 2012
Composite numbers n such that the number of digits of the period of 1/n divides n-1. - Davide Rotondo, Dec 16 2020

R. K. Guy, Unsolved Problems in Number Theory, A12.
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..1000
C. Pomerance & N. J. A. Sloane, Correspondence, 1991
Index entries for sequences related to pseudoprimes

Cf. A001567 (pseudoprimes to base 2), A005382, A121014, A121912.

Select[Range[15300], ! PrimeQ[ # ] && PowerMod[10, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

A057327 First member of a prime quadruple in a 2p-1 progression.

Original entry on oeis.org

1531, 2131, 2311, 3061, 6211, 6841, 7411, 10321, 13681, 15391, 16651, 18121, 22531, 23011, 24391, 29671, 30781, 31771, 33301, 35311, 41491, 44371, 46411, 54601, 56311, 57991, 60331, 61381, 66601, 67651, 78031, 83431, 85381, 88741

Offset: 1

Patrick De Geest, Aug 15 2000

nonn

Numbers n such that n remains prime through 3 iterations of function f(x) = 2x - 1.

			Quadruplets are (1531,3061,6121,12241), (2131,4261,8521,17041), ...

Cf. (A005382 and A005383), A057326, A057328, A057329, A057330, A005603.

Select[Prime[Range[9000]],And@@PrimeQ[NestList[2#-1&,#,3]]&] (* Harvey P. Dale, May 27 2012 *)

Offset changed by Andrew Howroyd, Aug 13 2024

A066071 Nonprime numbers k such that phi(k) + 1 is prime.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 18, 21, 22, 26, 27, 28, 32, 34, 36, 38, 40, 42, 46, 48, 49, 54, 55, 57, 58, 60, 62, 63, 74, 75, 76, 77, 82, 86, 88, 91, 93, 94, 95, 98, 99, 100, 106, 108, 110, 111, 114, 115, 117, 118, 119, 122, 124, 125, 126, 132, 133, 134, 135, 142, 145, 146

Offset: 1

Labos Elemer, Dec 03 2001

nonn

A039698 with the primes removed. For every prime p, 2p is in the sequence. - Ray Chandler, May 26 2008

Includes 3*p for p in A005382 and p^2 for p in A065508. - Robert Israel, Dec 29 2017

			Solutions to 1+phi(x)=13 are {13, 21, 26, 28, 36, 42} of which the 5 composites are in the sequence.

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A000010, A000040, A005382, A006093, A039649, A039698, A058339, A058340, A065508, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

[n: n in [1..200] |not IsPrime(n) and IsPrime(EulerPhi(n)+1)]; // Vincenzo Librandi, Jul 02 2016

select(n -> not isprime(n) and isprime(1+numtheory:-phi(n)), [$1..1000]); # Robert Israel, Dec 29 2017

Select[Complement[Range@ #, Prime@ Range@ PrimePi@ #] &@ 150, PrimeQ[EulerPhi@ # + 1] &] (* Michael De Vlieger, Jul 01 2016 *)

isok(k) = { !isprime(k) && isprime(eulerphi(k) + 1) } \\ Harry J. Smith, Nov 10 2009

A168421 Small Associated Ramanujan Prime, p_(i-n).

Original entry on oeis.org

2, 7, 11, 17, 23, 29, 31, 37, 37, 53, 53, 59, 67, 79, 79, 89, 97, 97, 127, 127, 127, 127, 127, 137, 137, 149, 157, 157, 179, 179, 191, 191, 211, 211, 211, 223, 223, 223, 233, 251, 251, 257, 293, 293, 307, 307, 307, 307, 307, 331, 331, 331

Offset: 1

John W. Nicholson, Nov 25 2009

nonn

a(n) is the smallest prime p_(k+1-n) on the left side of the Ramanujan Prime Corollary, 2*p_(i-n) > p_i for i > k, where the n-th Ramanujan Prime R_n is the k-th prime p_k. [Comment clarified and shortened by Jonathan Sondow, Dec 20 2013]
Smallest prime number, a(n), such that if x >= a(n), then there are at least n primes between x and 2x exclusively.
This is very useful in showing the number of primes in the range [p_k, 2*p_(i-n)] is greater than or equal to 1. By taking into account the size of the gaps between primes in [p_(i-n),p_k], one can see that the average prime gap is about log(p_k) using the following R_n / (2*n) ~ log(R_n).
Proof of Corollary: See Wikipedia link
The number of primes until the next Ramanujan prime, R_(n+1), can be found in A190874.
Not the same as A124136.
A084140(n) is the smallest integer where ceiling ((A104272(n)+1)/2), a(n) is the next prime after A084140(n). - John W. Nicholson, Oct 09 2013
If a(n) is in A005382(k) then A005383(k) is a twin prime with the Ramanujan prime, A104272(n) = A005383(k) - 2, and A005383(k) = A168425(n). If this sequence has an infinite number of terms in A005382, then the twin prime conjecture can be proved. - John W. Nicholson, Dec 05 2013
Except for A000101(1)=3 and A000101(2)=5, A000101(k) = a(n). Because of the large size of a gap, there are many repeats of the prime number in this sequence. - John W. Nicholson, Dec 10 2013
For some n and k, we see that a(n) = A104272(k) as to form a chain of primes similar to a Cunningham chain. For example (and the first example), a(2) = 7, links A104272(2) = 11 = a(3), links A104272(3) = 17 = a(4), links A104272(4) = 29 = a(6), links A104272(6) = 47. Note that the links do not have to be of a form like q = 2*p+1 or q = 2*p-1. - John W. Nicholson, Dec 14 2013
Srinivasan's Lemma (2014): p_(k-n) < (p_k)/2 if R_n = p_k and n > 1. Proof: By the minimality of R_n, the interval ((p_k)/2,p_k] contains exactly n primes, so p_(k-n) < (p_k)/2. - Jonathan Sondow, May 10 2014
In spite of the name Small Associated Ramanujan Prime, a(n) is not a Ramanujan prime for many values of n. - Jonathan Sondow, May 10 2014
Prime index of a(n), pi(a(n)) = i-n, is equal to A179196(n) - n + 1. - John W. Nicholson, Sep 15 2015
All maximal prime pairs in A002386 and A000101 are bounded by, for a particular n and i, the prime A104272(n) and twice a prime in A000040() following a(n). This means the gap between maximal prime pair cannot be more than twice the prior maximal prime gap. - John W. Nicholson, Feb 07 2019

			For n=10, the n-th Ramanujan prime is A104272(n)= 97, the value of k = 25, so i is >= 26, i-n >= 16, the i-n prime is 53, and 2*53 = 106. This leaves the range [97, 106] for the 26th prime which is 101. In this example, 53 is the small associated Ramanujan prime.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010; Amer. Math. Monthly 116 (2009) 630-635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2
Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19
Wikipedia, Ramanujan Prime

Cf. A000101, A005382, A005383, A084139, A084140.
Cf. A104272, A124136, A168425, A179196, A190874.
Cf. A165959 (range size), A230147 (records).

nn = 100;
t = Table[0, {nn}];
Do[m = PrimePi[2n] - PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15 nn}];
A084139 = Join[{1}, t];
a[n_] := NextPrime[A084139[[n]]];
Array[a, nn] (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A084139 *)

use ntheory ":all"; say next_prime((nth_ramanujan_prime($)+1) >> 1) for 1..100; # _Dana Jacobsen, Mar 02 2016

a(n) = prime(primepi(A104272(n)) + 1 - n).
a(n) = nextprime(A084139(n+1)), where nextprime(x) is the next prime > x. Note: some A084139(n) may be prime, therefore nextprime(x) not equal to x. - John W. Nicholson, Oct 11 2013
a(n) = nextprime(A084140(n)). - John W. Nicholson, Oct 11 2013

Extended by T. D. Noe, Nov 22 2010

A158015 Primes p such that 6*p-1 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 23, 29, 43, 47, 53, 59, 67, 103, 107, 109, 113, 127, 137, 157, 163, 197, 199, 227, 229, 239, 269, 283, 313, 317, 347, 359, 373, 379, 383, 389, 397, 439, 443, 449, 457, 463, 467, 523, 569, 577, 593, 599, 613, 617, 647, 653, 709, 733, 743, 773

Offset: 1

Roger L. Bagula, Mar 11 2009

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

Cf. A005382 for the type 2p-1, A062737 for 4p-1, A158016 for 8p-1, A158017 for 10p-1.

Primes in A024898, i.e., intersection of A024898 with A000040.

[p: p in PrimesUpTo(800) | IsPrime(6*p-1)]; // Vincenzo Librandi, Apr 14 2013

Select[Prime[Range[200]], PrimeQ[(6 # - 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

Edited by the Associate Editors of the OEIS, Apr 22 2009

A057329 First member of a prime sextuplet in a 2p-1 progression.

Original entry on oeis.org

16651, 33301, 165901, 331801, 385591, 445741, 478801, 580471, 1203121, 1768441, 1943371, 2041201, 2131141, 2240941, 2340661, 2393431, 2526721, 3277471, 3536881, 3623881, 3880381, 3897631, 4123621, 4415371, 4481881, 5278591

Offset: 1

Patrick De Geest, Aug 15 2000

nonn

Numbers n such that n remains prime through 5 iterations of function f(x) = 2x - 1.

			First sextuplet is (16651,33301,66601,133201,266401,532801).

Index entries for sequences related to primes in arithmetic progressions

Cf. A005382 and A005383; A057326, A057327, A057328, A057330, A005603.

[ p: p in PrimesUpTo(6*10^6) | forall{q: k in [1..5] | IsPrime(q) where q is 2^k*(p-1)+1} ];  // Bruno Berselli, Nov 23 2011

Offset changed by Andrew Howroyd, Aug 13 2024

A062737 Primes p such that 4p-1 is also prime.

Original entry on oeis.org

2, 3, 5, 11, 17, 41, 53, 71, 83, 131, 137, 173, 197, 227, 263, 281, 293, 383, 431, 467, 503, 521, 563, 587, 593, 617, 677, 683, 701, 743, 797, 827, 887, 911, 977, 983, 1013, 1061, 1091, 1151, 1163, 1181, 1277, 1307, 1361, 1481, 1511, 1523, 1553, 1607, 1613

Offset: 1

Reiner Martin, Jul 12 2001

For n>1, 4*a(n) is a solution to the equation phi(x-1) - phi(x) = x/2. - Farideh Firoozbakht, Dec 17 2014

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A005384, A007700, A005382.

select(n -> isprime(n) and isprime(4*n-1), [2,seq(2*k+1,k=1..1000)]); # Robert Israel, Dec 17 2014

{ n=0; forprime (p=2, 10^5, if (isprime(4*p - 1), write("b062737.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 10 2009

A020138 Pseudoprimes to base 9.

Original entry on oeis.org

4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288, 1387, 1541, 1729, 1891, 2465, 2501, 2665, 2701, 2806, 2821, 2926, 3052, 3281, 3367, 3751, 4376, 4636, 4961, 5356, 5551, 6364, 6601, 6643, 7081, 7381, 7913, 8401

Offset: 1

David W. Wilson

nonn

This sequence is a subsequence of A122786. In fact the terms are composite terms n of A122786 such that gcd(n,3)=1. Theorem: If both numbers q & 2q-1 are primes greater than 3 and n=q*(2q-1) then 9^(n-1)==1 (mod n) (n is in the sequence). So for n>2 A005382(n)* (2*A005382(n)-1) is in the sequence; 91,703,1891,2701,12403,18721,... is the related subsequence. - Farideh Firoozbakht, Sep 15 2006

Composite numbers n such that 9^(n-1) == 1 (mod n).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..159 from R. J. Mathar, terms 160..1000 from T. D. Noe)
Index entries for sequences related to pseudoprimes

Cf. A001567 (pseudoprimes to base 2), A005382, A122786.

Select[Range[8500], ! PrimeQ[ # ] && PowerMod[9, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

A077068 Semiprimes of the form prime + 1.

Original entry on oeis.org

4, 6, 14, 38, 62, 74, 158, 194, 278, 314, 398, 422, 458, 542, 614, 662, 674, 734, 758, 878, 998, 1094, 1154, 1202, 1214, 1238, 1322, 1382, 1454, 1622, 1658, 1754, 1874, 1934, 1994, 2018, 2138, 2342, 2474, 2558, 2594, 2798, 2858, 2918, 3062, 3218, 3254

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

a(n) = A005383(n)+1 = 2*A005382(n).
There are 672 semiprimes of form prime+1 below 100000.
a(n) = A232342(n) + A077065(n). - Reinhard Zumkeller, Dec 16 2013

			A001358(25)=74=2*37 is a term as 74=A000040(21)+1=73+1.

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

Cf. A008864, A001358, A000040, A077065, A077067.

a077068 n = a077068_list !! (n-1)
a077068_list = filter ((== 1) . a010051 . (`div` 2)) a008864_list
-- Reinhard Zumkeller, Nov 22 2013

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#-1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[ Prime@ Range@ 460 +1,PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x+1|x<-primes(10^5),bigomega(x+1)==2] \\ Charles R Greathouse IV, Nov 22 2013

is(n)=isprime(n-1) && isprime(n\2) \\ Charles R Greathouse IV, Mar 20 2016

A010051(A008864(n)/2) = A064911(A008864(n)) = 1. - Reinhard Zumkeller, Nov 22 2013

cp's OEIS Frontend

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

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A005939 Pseudoprimes to base 10.

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A057327 First member of a prime quadruple in a 2p-1 progression.

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Mathematica

Extensions

A066071 Nonprime numbers k such that phi(k) + 1 is prime.

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Magma

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Mathematica

PARI

A168421 Small Associated Ramanujan Prime, p_(i-n).

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Mathematica

Perl

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A158015 Primes p such that 6*p-1 is also prime.

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Magma

Mathematica

Extensions

A057329 First member of a prime sextuplet in a 2p-1 progression.

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