A005382 -id:A005382 - cp's OEIS frontend

A063908 Numbers k such that k and 2*k-3 are primes.

3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 53, 67, 71, 83, 97, 101, 107, 113, 127, 137, 157, 167, 181, 191, 193, 211, 223, 233, 241, 251, 263, 283, 311, 317, 331, 347, 373, 421, 431, 433, 443, 457, 461, 487, 521, 547, 563, 577, 587, 613, 617, 631, 641, 643, 647

Offset: 1

N. J. A. Sloane, Aug 31 2001

nonn

If p is in this sequence then the products of positive powers of 3, p and 2p-3 are entries in A086486. - Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003
Median prime of AP3's starting at 3, i.e., triples of primes (3,p,q) in arithmetic progression. - M. F. Hasler, Sep 24 2009
a(n) = sum of the coprimes(p) mod (p+1). - J. M. Bergot, Nov 13 2014
A010051(2*a(n)-3) = 1. - Reinhard Zumkeller, Jul 02 2015
A098090 INTERSECT A000040. - R. J. Mathar, Mar 23 2017

			From _K. D. Bajpai_, Nov 29 2019: (Start)
a(5) = 13 is prime and 2*13 - 3 = 23 is also prime.
a(6) = 17 is prime and 2*17 - 3 = 31 is also prime.
(End)

Harry J. Smith and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Carlos Rivera, Puzzle 34.- Prime Triplets in arithmetic progression, The Prime Puzzles & Problems Connection. [From _M. F. Hasler_, Sep 24 2009]

Cf. A005382.
Cf. A000040, A010051, A088878, A172287.
Cf. A259730.

a063908 n = a063908_list !! (n-1)
a063908_list = filter
   ((== 1) . a010051' . (subtract 3) . (* 2)) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

[n : n in [0..700] | IsPrime(n) and IsPrime(2*n-3)]; // Vincenzo Librandi, Nov 14 2014

select(k -> andmap(isprime, [k, 2*k-3]), [seq(k, k=1.. 10^4)]); # K. D. Bajpai, Nov 29 2019

Select[Prime[Range[6! ]],PrimeQ[2*#-3]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

{ n=0; p=1; for (m=1, 10^9, p=nextprime(p+1); if (isprime(2*p - 3), write("b063908.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 02 2009

forprime( p=1,default(primelimit), isprime(2*p-3) && print1(p",")) \\ M. F. Hasler, Sep 24 2009

a(n) = A241817(n)/2. - Wesley Ivan Hurt, Apr 08 2018

A129521 Numbers of the form pq, p and q prime with q=2p-1.

Original entry on oeis.org

6, 15, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381

Offset: 1

Reinhard Zumkeller, Apr 19 2007

nonn

All terms are Fermat 4-pseudoprimes, i.e., satisfy 4^n == 4 (mod n). See A020136 and A122781.

David W. Wilson, Table of n, a(n) for n = 1..1000
D. E. Iannucci, When the small divisors of a natural number are in arithmetic progression, INTEGERS, Electronic Journal of Combinatorial Number Theory, #77, 2018. See p. 9.

Cf. A005382, A005383.
Subsequence of A006881, A129510, and A122781.
Intersection of A000384 and A001358, "hexagonal semiprimes". - Wesley Ivan Hurt, Jul 04 2013
Cf. A141768, A191311, A191592, A245365, A259676, A259677.

a129521 n = p * (2 * p - 1) where p = a005382 n
-- Reinhard Zumkeller, Nov 10 2013

[2*n^2-n: n in [0..1000]|IsPrime(n) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 27 2010

p = Select[Prime[Range[155]], PrimeQ[2# - 1] &]; p (2p - 1) (* Robert G. Wilson v, Sep 11 2011 *)

forprime(p=2,10000,q=2*p-1;if(isprime(q),print1(p*q,", ")))

a(n) = A005382(n)*A005383(n).

A057330 First member of a prime n-tuplet in a 2p-1 progression.

Original entry on oeis.org

2, 2, 2, 1531, 1531, 16651, 16651, 15514861, 857095381, 205528443121, 1389122693971, 216857744866621, 758083947856951, 69257563144280941, 69257563144280941, 3203000719597029781

Offset: 1

Patrick De Geest, Aug 15 2000

Initial terms of A000040, A005382, A057326, A057327, A057328, A057329.

Dirk Augustin, Smallest existing Cunningham Chains of given length
Chris Caldwell, The Prime Glossary: Cunningham chain
Tony Forbes, Cunningham Chain of length 16, Dec 05 1997
Index entries for sequences related to primes in arithmetic progressions

A005603 Smallest prime beginning a complete Cunningham chain (of the second kind) of length n.

Original entry on oeis.org

11, 7, 2, 2131, 1531, 385591, 16651, 15514861, 857095381, 205528443121, 1389122693971, 216857744866621, 758083947856951, 107588900851484911, 69257563144280941, 3203000719597029781

Offset: 1

N. J. A. Sloane

The chain begins with a prime number p; next term p' (a prime) is produced forming 2p-1; next term p"=2p'-1, etc. "Complete" means that each chain is exactly n primes long (i.e. the chain cannot be a subchain of another one). That is why this sequence is slightly different from A064812, where the 6th term (33301) is smaller than here (385591) but is the second one of a seven primes sequence and therefore doesn't *start* a sequence.

According to Augustin's web site, the numbers 107588900851484911, 69257563144280941, 3203000719597029781 are also in the sequence. - Dmitry Kamenetsky, May 14 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Dirk Augustin, Cunningham Chain records.
C. K. Caldwell, Cunningham chain.
G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.

See A064812 for another version.

Cf. (A005382 and A005383), A057326, A057327, A057328, A057329, A057330, A057331, A005602.

6th term corrected from 385591 on Feb 23 1995, at Robert G. Wilson v's suggestion
a(14) and a(15) found by Paul Jobling (Paul.Jobling(AT)WhiteCross.com) [Oct 23 2000]
a(6) reverted to original value by Sean A. Irvine, Jul 10 2016
a(16) from Augustin's page, comment corrected by Jens Kruse Andersen, Jun 14 2014
Edited by N. J. A. Sloane, Nov 03 2018 at the suggestion of Georg Fischer, Nov 03 2018, merging a duplicate entry with this one.
In Augustin's web page there are 7 or so more terms which could be added here, or alternatively used to create a b-file. - Georg Fischer, Nov 03 2018

A068443 Triangular numbers which are the product of two primes.

Original entry on oeis.org

6, 10, 15, 21, 55, 91, 253, 703, 1081, 1711, 1891, 2701, 3403, 5671, 12403, 13861, 15931, 18721, 25651, 34453, 38503, 49141, 60031, 64261, 73153, 79003, 88831, 104653, 108811, 114481, 126253, 146611, 158203, 171991, 188191, 218791, 226801, 258121, 269011

Offset: 1

Stephan Wagler (stephanwagler(AT)aol.com), Mar 09 2002

These triangular numbers are equal to p * (2p +- 1).
All terms belong to A006987. For n>2 all terms are odd and belong to A095147. - Alexander Adamchuk, Oct 31 2006
A156592 is a subsequence. - Reinhard Zumkeller, Feb 10 2009
Triangular numbers with exactly 4 divisors. - Jon E. Schoenfield, Sep 05 2018

			Triangular numbers begin 0, 1, 3, 6, 10, ...; 6=2*3, and 2 and 3 are two distinct primes; 10=2*5, and 2 and 5 are two distinct primes, etc. - _Vladimir Joseph Stephan Orlovsky_, Feb 27 2009
a(11) = 1891 and 1891 = 31 * 61.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000217, A005382, A005384, A006987, A095147, A001358, A005385, A006881, A007304, A066179, A111206, A157342, A157344-A157347, A157352-A157357, A164977.

Cf. A075875.

q:= n-> is(numtheory[bigomega](n)=2):
select(q, [i*(i+1)/2$i=0..1000])[];  # Alois P. Heinz, Mar 27 2024

Select[ Table[ n(n + 1)/2, {n, 1000}], Apply[Plus, Transpose[ FactorInteger[ # ]] [[2]]] == 2 &]
Select[Accumulate[Range[1000]],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 03 2016 *)

list(lim)=my(v=List());forprime(p=2,(sqrtint(lim\1*8+1)+1)\4, if(isprime(2*p-1),listput(v,2*p^2-p)); if(isprime(2*p+1), listput(v,2*p^2+p))); Vec(v) \\ Charles R Greathouse IV, Jun 13 2013

A010054(a(n))*A064911(a(n)) = 1. - Reinhard Zumkeller, Dec 03 2009

a(n) = A000217(A164977(n)). - Zak Seidov, Feb 16 2015

Edited by Robert G. Wilson v, Jul 08 2002

Definition corrected by Zak Seidov, Mar 09 2008

A005936 Pseudoprimes to base 5.

Original entry on oeis.org

4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881, 11041, 11476, 12801, 13021, 13333, 13981, 14981, 15751, 15841, 16297, 17767, 21361, 22791, 23653, 24211, 25327, 25351, 29341, 29539

Offset: 1

N. J. A. Sloane

nonn

According to Karsten Meyer, 4 should be excluded, following the strict definition in Crandall and Pomerance. - May 16 2006
Theorem: If both numbers q and (2q - 1) are primes (q is in the sequence A005382) then n = q*(2q - 1) is a pseudoprime to base 5 (n is in the sequence) if and only if q is of the form 10k + 1. 1891, 88831, 146611, 218791, 721801, ... are such terms. This sequence is a subsequence of A122782. - Farideh Firoozbakht, Sep 14 2006
Composite numbers n such that 5^(n-1) == 1 (mod n).

R. Crandall and C. Pomerance, "Prime Numbers - A Computational Perspective", Second Edition, Springer Verlag 2005, ISBN 0-387-25282-7 Page 132 (Theorem 3.4.2. and Algorithm 3.4.3)
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 124, p. 43, Ellipses, Paris 2008.
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).

R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..92893 (terms a(1)-a(776) from R. J. Mathar, a(777)-a(1000) from T. D. Noe)
J. Bernheiden, Pseudoprimes (Text in German)
C. Pomerance & N. J. A. Sloane, Correspondence, 1991
F. Richman, Primality testing with Fermat's little theorem
Eric Weisstein's World of Mathematics, Fermat Pseudoprime
Index entries for sequences related to pseudoprimes

Pseudoprimes to other bases: A001567 (2), A005935 (3), A005937 (6), A005938 (7), A005939 (10).

Cf. A005382, A122782.

base = 5; t = {}; n = 1; While[Length[t] < 100, n++; If[! PrimeQ[n] && PowerMod[base, n-1, n] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Feb 21 2012 *)
Select[Range[30000],CompositeQ[#]&&PowerMod[5,#-1,#]==1&] (* Harvey P. Dale, Jul 21 2023 *)

More terms from David W. Wilson, Aug 15 1996

A005938 Pseudoprimes to base 7.

Original entry on oeis.org

6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, 10225, 10585, 10621, 11041, 11521, 12025, 13665, 14089, 16725, 16806, 18721, 19345, 20197, 20417, 20425, 22945, 25829, 26419, 29234, 29341, 29857, 29891, 30025, 30811, 33227

Offset: 1

N. J. A. Sloane

nonn

According to Karsten Meyer, May 16 2006, 6 should be excluded, following the strict definition in Crandall and Pomerance.
Theorem: If both numbers q & 2q-1 are primes(q is in the sequence A005382) and n=q*(2q-1) then 7^(n-1)==1 (mod 7)(n is in the sequence) iff q=2 or mod(q,14) is in the set {1, 5, 13}. 6,703,18721,38503,88831,104653,146611,188191,... are such terms. This sequence is a subsequence of A122784. - Farideh Firoozbakht, Sep 14 2006
Composite numbers n such that 7^(n-1) == 1 (mod n).

R. Crandall and C. Pomerance, "Prime Numbers - A Computational Perspective", Second Edition, Springer Verlag 2005, ISBN 0-387-25282-7 Page 132 (Theorem 3.4.2. and Algorithm 3.4.3)
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).

R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..87448 (terms a(1)-a(697) from R. J. Mathar, a(698)-a(1000) from T. D. Noe)
J. Bernheiden, Pseudoprimes (Text in German)
C. Pomerance & N. J. A. Sloane, Correspondence, 1991
F. Richman, Primality testing with Fermat's little theorem
Index entries for sequences related to pseudoprimes

Pseudoprimes to other bases: A001567 (2), A005935 (3), A005936 (5), A005937 (6), A005939 (10).

Cf. A005382, A122784.

Select[Range[31000], ! PrimeQ[ # ] && PowerMod[7, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 14 2006 *)

from sympy import isprime
def ok(n): return pow(7, n-1, n) == 1 and not isprime(n)
print(list(filter(ok, range(1, 34000)))) # Michael S. Branicky, Jun 25 2021

A020136 Fermat pseudoprimes to base 4.

Original entry on oeis.org

15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071, 2465, 2701, 2821, 3133, 3277, 3367, 3683, 4033, 4369, 4371, 4681, 4795, 4859, 5461, 5551, 6601, 6643, 7957, 8321, 8481, 8695, 8911, 9061, 9131

Offset: 1

David W. Wilson

nonn

If q and 2q-1 are odd primes, then n=q*(2q-1) is in the sequence. So for n>1, A005382(n)*(2*A005382(n)-1) form a subsequence (cf. A129521). - Farideh Firoozbakht, Sep 12 2006
Primes q and 2q-1 are a Cunningham chain of the second kind. - Walter Nissen, Sep 07 2009
Composite numbers n such that 4^(n-1) == 1 (mod n). - Michel Lagneau, Feb 18 2012

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Chris Caldwell, Cunningham chain.
Chris Caldwell et al., Top Twenty Cunningham Chains (2nd kind).
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes

Subsequence of A122781.
Contains A001567 (Fermat pseudoprimes to base 2) as a subsequence.
Cf. A005382, A129521.

Select[Range[9200], ! PrimeQ[ # ] && PowerMod[4, # - 1, # ] == 1 &] (* Farideh Firoozbakht, Sep 12 2006 *)

isok(n) = (Mod(4, n)^(n-1)==1) && !isprime(n) && (n>1); \\ Michel Marcus, Apr 27 2018

A023509 Greatest prime divisor of prime(n) + 1.

Original entry on oeis.org

3, 2, 3, 2, 3, 7, 3, 5, 3, 5, 2, 19, 7, 11, 3, 3, 5, 31, 17, 3, 37, 5, 7, 5, 7, 17, 13, 3, 11, 19, 2, 11, 23, 7, 5, 19, 79, 41, 7, 29, 5, 13, 3, 97, 11, 5, 53, 7, 19, 23, 13, 5, 11, 7, 43, 11, 5, 17, 139, 47, 71, 7, 11, 13, 157, 53, 83, 13, 29, 7, 59, 5, 23, 17, 19, 3, 13

Offset: 1

Clark Kimberling

nonn

A005382 is the records subsequence of this sequence. - David James Sycamore, May 05 2025

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A006530, A008864, A005382.

Table[FactorInteger[Prime[n] + 1][[-1, 1]], {n, 100}]

A023509(n) = {local(f);f=factor(prime(n)+1);f[matsize(f)[1],1]} \\ Michael B. Porter, Feb 02 2010

a(n) = A006530(A008864(n)). - R. J. Mathar, Feb 06 2019

A057328 First member of a prime 5-tuple in a 2p-1 progression.

Original entry on oeis.org

1531, 6841, 15391, 16651, 33301, 44371, 57991, 66601, 83431, 105871, 145021, 150151, 165901, 199621, 209431, 212851, 231241, 242551, 291271, 319681, 331801, 346141, 377491, 381631, 385591, 445741, 451411, 478801, 481021, 506791, 507781

Offset: 1

Patrick De Geest, Aug 15 2000

nonn

Numbers n such that n remains prime through 4 iterations of function f(x) = 2x - 1.

			Quintuplets are (1531, 3061, 6121, 12241, 24481), (6841, 13681, 27361, 54721, 109441), ...

Cf. A005382, A005383, A057326, A057327, A057329, A057330, A005603.

[ p: p in PrimesUpTo(6*10^5) | forall{q: k in [1..4] | IsPrime(q) where q is 2^k*(p-1)+1} ];  // Bruno Berselli, Nov 23 2011

pQ[n_] := And @@ PrimeQ[NestList[2 # - 1 &, n, 4]]; t = {}; Do[p = Prime[n]; If[pQ[p], AppendTo[t, p]], {n, 42500}]; t (* Jayanta Basu, Jun 17 2013 *)
Select[Prime[Range[50000]],AllTrue[Rest[NestList[2#-1&,#,4]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 30 2019 *)

cp's OEIS Frontend

A063908 Numbers k such that k and 2*k-3 are primes.

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A129521 Numbers of the form p*q, p and q prime with q=2*p-1.

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A057330 First member of a prime n-tuplet in a 2p-1 progression.

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A005603 Smallest prime beginning a complete Cunningham chain (of the second kind) of length n.

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A068443 Triangular numbers which are the product of two primes.

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A005936 Pseudoprimes to base 5.

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A005938 Pseudoprimes to base 7.

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A129521 Numbers of the form pq, p and q prime with q=2p-1.