A005382 -id:A005382 - cp's OEIS frontend

A137288 Numbers k such that 2*prime(k)-1 is prime.

1, 2, 4, 8, 11, 12, 22, 25, 34, 37, 46, 47, 50, 58, 63, 67, 68, 73, 75, 85, 95, 101, 106, 110, 111, 114, 121, 125, 129, 141, 145, 151, 159, 163, 168, 169, 180, 193, 203, 207, 211, 222, 226, 232, 242, 254, 258, 260, 274, 285

Offset: 1

Ctibor O. Zizka, Apr 05 2008

			k=11: 2*prime(11) - 1 = 2*31 - 1 = 61 is prime, so k=11 belongs to the sequence.

Carl R. White, Table of n, a(n) for n = 1..10000

Cf. A000040, A005382 (corresponding primes).

Select[Range[285],PrimeQ[2Prime[#]-1]&] (* James C. McMahon, May 22 2025 *)

More terms from Paolo P. Lava, Apr 15 2008

A105657 Numbers n such that p1=2n+3, p2=4n+5, p3=6n+7, p4=8n+9, p5=10n+11, p6=12n+13, p7=14n+15 and p8=16n+17 are all prime.

Original entry on oeis.org

256409, 11120339, 13243229, 49798979, 296504669, 510578774, 520649219, 640598279, 674992499, 713074004, 830453714, 947378984

Offset: 1

Zak Seidov, Apr 16 2005

nonn

Cf. A005382, A005383, A105610, A105652 - A105656.

A109998 Non-Cunningham primes: primes isolated from any Cunningham chain under any iteration of 2p+-1 or (p+-1)/2.

Original entry on oeis.org

17, 43, 67, 71, 101, 103, 109, 127, 137, 149, 151, 163, 181, 197, 223, 241, 257, 269, 283, 311, 317, 349, 353, 373, 389, 401, 409, 433, 449, 461, 463, 487, 521, 523, 557, 569, 571, 599, 617, 631, 643, 647, 677, 701, 709, 739, 751, 769, 773, 787, 797, 821

Offset: 1

Alexandre Wajnberg, Sep 01 2005

The condition that neither 2p - 1 nor 2p + 1 be prime is equivalent to ((p-1) mod 3 = 0) or ((p+1) mod 3 = 0). For example, the prime p = 2^607 - 1 is not in this sequence because p + 1 mod 3 = 2. - Washington Bomfim, Oct 30 2009

			a(1) = 17 is here because 17 * 2 + 1 = 35, 17 * 2 - 1 = 33; (17+1)/2 = 9, (17-1)/2 = 8: four composite numbers.

Chris Caldwell's Prime Glossary, Cunningham chains.
Douglas S. Stones, On prime chains, arXiv:0908.2166 [math.NT] [From _Washington Bomfim_, Oct 30 2009, edited by _R. J. Mathar_, Mar 01 2010]

Cf. A005385, A005383, A060254, A005384, A005382, A068497, A059455, A059762, A057326, A023272, A023302, A059764, A057328, A023330, A005602, A064812, A005603.

nonCunninghamPrimes = {}; Do[p = Prime[n]; If[!PrimeQ[2p - 1] && !PrimeQ[2p + 1] && !PrimeQ[(p - 1)/2] && !PrimeQ[(p + 1)/2], AppendTo[nonCunninghamPrimes, p]], {n, 6!}]; nonCunninghamPrimes (* Vladimir Joseph Stephan Orlovsky, Mar 22 2009 *)

Corrected and extended by Ray Chandler, Sep 02 2005

Replaced link to cached arXiv URL with link to the abstract - R. J. Mathar, Mar 01 2010

A121014 Nonprime terms in A121912.

Original entry on oeis.org

1, 6, 9, 10, 15, 18, 30, 33, 45, 55, 90, 91, 99, 165, 246, 259, 370, 385, 451, 481, 495, 505, 561, 657, 703, 715, 909, 1035, 1045, 1105, 1233, 1626, 1729, 2035, 2409, 2465, 2821, 2981, 3333, 3367, 3585, 4005, 4141, 4187, 4521, 4545, 5005, 5461, 6533, 6541

Offset: 1

N. J. A. Sloane, Sep 06 2006

Theorem: If both numbers q and 2q-1 are primes (q is in the sequence A005382) and n=q*(2q-1) then 10^n == 10 (mod n) (n is in the sequence A121014) iff q<5 or mod(q, 20) is in the set {1, 7, 19}. 6,15,91,703,12403,38503,79003,188191,269011,... are such terms. A005939 is a subsequence of this sequence. - Farideh Firoozbakht, Sep 15 2006

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

Cf. A005382, A005939, A121014, A121912.

Select[Range[10^4], ! PrimeQ[ # ] && PowerMod[10, #, # ] == Mod[10, # ] &] (* Ray Chandler, Sep 06 2006 *)

for(n=1,7000,if(!isprime(n),k=10^n;if((k-10)%n==0,print1(n,",")))) \\ Klaus Brockhaus, Sep 06 2006

Theorem: If both numbers q and 2q-1 are primes and n=q*(2q-1) then 10^n == 10 (mod n) (n is in the sequence) iff q<5 or mod(q, 20) is in the set {1, 7, 19}. - Farideh Firoozbakht, Sep 11 2006

Extended by Ray Chandler and Klaus Brockhaus, Sep 06 2006

A122786 Nonprimes n such that 9^n == 9 (mod n).

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 15, 18, 24, 28, 36, 45, 52, 66, 72, 91, 121, 153, 205, 276, 286, 364, 366, 369, 396, 435, 511, 532, 561, 616, 671, 697, 703, 726, 804, 946, 949, 1035, 1036, 1105, 1128, 1288, 1387, 1541, 1729, 1737, 1845, 1854, 1891, 2196, 2465, 2501, 2556, 2665

Offset: 1

Farideh Firoozbakht, Sep 12 2006

nonn

Theorem: If both numbers q and 2q-1 are primes and n=q*(2q-1) then 9^n==9 (mod n) (n is in the sequence). So A005382*(2*A005382-1)= 6,15,91,703,1891,2701,12403,18721,... is the related subsequence. A020138 is a subsequence of this sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A005382, A020138.

q:= n-> is(not isprime(n) and (9 &^ n mod n) = (9 mod n)):
select(q, [$1..3000])[];  # Alois P. Heinz, Mar 06 2019

Select[Range[4000], ! PrimeQ[ # ] && Mod[9^#, # ] == Mod[9, # ] &]
Join[{1,4,6,8,9},Select[Range[3000],CompositeQ[#]&&PowerMod[9,#,#]==9&]] (* Harvey P. Dale, Jul 17 2014 *)

isok(n) = !isprime(n) && (Mod(9,n)^n == Mod(9, n)); \\ Michel Marcus, Mar 06 2019

A158017 Primes p such that 10*p-1 is also prime.

Original entry on oeis.org

2, 3, 11, 23, 41, 71, 83, 101, 107, 113, 149, 167, 179, 227, 239, 269, 311, 317, 347, 353, 389, 479, 491, 521, 557, 569, 587, 647, 653, 683, 809, 821, 827, 839, 863, 911, 977, 983, 1091, 1229, 1259, 1283, 1289, 1301, 1367, 1373, 1439, 1487, 1493, 1607, 1619

Offset: 1

Roger L. Bagula, Mar 11 2009

The family of prime sequences that generate primes k*p-1 for k = 2, 4, 6, 8, ... also comprises A005382 (k=2), A062737 (k=4), A158015 (k=6), and A158016 (k=8).

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

Cf. A005382, A062737.

[p: p in PrimesUpTo(3000)|IsPrime(10*p-1)] // Vincenzo Librandi, Jan 29 2011

Flatten[Table[If[PrimeQ[n] && PrimeQ[10*n - 1], n, {}], {n, 1, 10000}]]
Select[Prime[Range[600]], PrimeQ[(10 # - 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

A171517 Primes p such that 2*p+11 is prime.

Original entry on oeis.org

3, 13, 31, 43, 73, 109, 151, 163, 181, 193, 199, 211, 223, 283, 331, 349, 373, 379, 409, 421, 433, 463, 499, 541, 571, 601, 613, 619, 643, 709, 739, 769, 823, 829, 883, 991, 1009, 1021, 1039, 1051, 1063, 1129, 1213, 1231, 1291, 1303, 1423, 1453, 1471, 1549

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2009

			2*3+11=17, which is prime.

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

Cf. A000040, A005382, A005384, A023204 - A023207, A063908 - A063912.

[p: p in PrimesUpTo(1600) | IsPrime(2*p+11)]; // Vincenzo Librandi, Apr 27 2014

Select[Prime[Range[6! ]],PrimeQ[2*#+11]&]

A286257 Compound filter: a(n) = P(A046523(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 5, 14, 12, 27, 5, 86, 14, 27, 23, 90, 12, 84, 27, 152, 23, 148, 5, 148, 27, 27, 80, 324, 25, 61, 44, 148, 23, 495, 5, 935, 61, 27, 61, 702, 5, 142, 61, 324, 138, 495, 23, 148, 90, 61, 23, 1426, 14, 265, 27, 90, 467, 324, 27, 430, 27, 61, 80, 2140, 12, 61, 183, 2144, 61, 495, 23, 607, 27, 495, 23, 2998, 23, 142, 90, 90, 142, 625, 5, 1426, 226, 27, 467

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A278223, A286255, A286256, A286258.

Cf. A005382 (gives the positions of 5's), A067756 (of 12's), A234098 (of 23's).

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286257(n) = (1/2)*(2 + ((A046523(n)+A046523((2*n)-1))^2) - A046523(n) - 3*A046523((2*n)-1));
for(n=1, 10000, write("b286257.txt", n, " ", A286257(n)));

from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a046523(n), a046523(2*n - 1)) # Indranil Ghosh, May 07 2017

(define (A286257 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ -1 n n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ -1 n n)))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A046523((2*n)-1))^2) - A046523(n) - 3*A046523((2*n)-1)).

a(n) = (1/2)*(2 + ((A046523(n)+A278223(n))^2) - A046523(n) - 3*A278223(n)).

A298758 Numbers k such that both k and 2k-1 are Poulet numbers (Fermat pseudoprimes to base 2).

Original entry on oeis.org

15709, 65281, 20770621, 104484601, 112037185, 196049701, 425967301, 2593182901, 16923897871, 32548281361, 45812984491, 52035130951, 55897227751, 82907336737, 90003640021, 92010062101, 138016057141, 204082130071, 310026150211, 620006892121, 622333751509

Offset: 1

Amiram Eldar, Jan 26 2018

nonn

2*a(n) - 1 = A303531(n) belongs to A217465. - Max Alekseyev, Apr 24 2018
Numbers k such that both k and 2k+1 are Poulet numbers are listed in A303447.
If p is a prime such that 2*p-1 is also a prime (A005382) and k = (2^(2*p-1)+1)/3 and 2*k-1 are both composites, then k is a term of this sequence (Rotkiewicz, 2000). - Amiram Eldar, Nov 09 2023

Amiram Eldar, Table of n, a(n) for n = 1..3031 (terms below 2^64)
Andrzej Rotkiewicz, Arithmetic progressions formed by pseudoprimes, Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 8, No. 1 (2000), pp. 61-74.

Subsequence of A001567.

Cf. A005382, A217465, A303447, A303448.

s = Import["b001567.txt", "Data"][[All, -1]]; n = Length[s];
aQ[n_] := ! PrimeQ[n] && PowerMod[2, (n - 1), n] == 1;
a = {}; Do[p = 2*s[[k]] - 1; If[aQ[p], AppendTo[a, s[[k]]]], {k, 1, n}]; a (* using the b-File from A001567 *)

isP(n) = (Mod(2, n)^n==2) && !isprime(n) && (n>1);
isok(n) = isP(n) && isP(2*n-1); \\ Michel Marcus, Mar 09 2018

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

cp's OEIS Frontend

A137288 Numbers k such that 2*prime(k)-1 is prime.

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A105657 Numbers n such that p1=2n+3, p2=4n+5, p3=6n+7, p4=8n+9, p5=10n+11, p6=12n+13, p7=14n+15 and p8=16n+17 are all prime.

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A109998 Non-Cunningham primes: primes isolated from any Cunningham chain under any iteration of 2p+-1 or (p+-1)/2.

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A121014 Nonprime terms in A121912.

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A122786 Nonprimes n such that 9^n == 9 (mod n).

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A158017 Primes p such that 10*p-1 is also prime.

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Magma

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A171517 Primes p such that 2*p+11 is prime.

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Magma

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A286257 Compound filter: a(n) = P(A046523(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

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