A005385 -id:A005385 - cp's OEIS frontend

A007700 Numbers n such that n, 2n+1, and 4n+3 all prime.

2, 5, 11, 41, 89, 179, 359, 509, 719, 1019, 1031, 1229, 1409, 1451, 1481, 1511, 1811, 1889, 1901, 1931, 2459, 2699, 2819, 3449, 3491, 3539, 3821, 3911, 5081, 5399, 5441, 5849, 6101, 6131, 6449, 7079, 7151, 7349, 7901, 8969, 9221, 10589, 10691, 10709, 11171

Offset: 1

N. J. A. Sloane, Simon Plouffe

nonn

The corresponding primes 2n+1 and 4n+3 respectively have n-1 and 2n primitive roots. - Lekraj Beedassy, Jan 07 2005

At n > 2, a(n) == {11,29} (mod 30). - Zak Seidov, Jan 31 2013

T. Moreau, personal communication.
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..10000
L. Blum, M. Blum, and M. Shub, A simple unpredictable pseudorandom number generator, SIAM J. Comput. 15 (1986), no. 2, 364-383.

Intersection of A005384 and A023213.

Cf. A005385, A023272, A023302, A023330, A057331, A005602.

A007700 := proc(n) local p1,p2; p1 := 2*n+1; p2 := 2*p1+1; if isprime(n) = true and isprime(p1)=true and isprime(p2)=true then RETURN(n); fi; end;

Select[Range[10^3*3], PrimeQ[ # ]&&PrimeQ[2*#+1]&&PrimeQ[4*#+3] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[1500]],AllTrue[{2#+1,4#+3},PrimeQ]&] (* Harvey P. Dale, Apr 12 2022 *)

is(n)=isprime(n)&&isprime(2*n+1)&&isprime(4*n+3) \\ Charles R Greathouse IV, Mar 21 2013

A005602 Smallest prime beginning a complete Cunningham chain of length n (of the first kind).

Original entry on oeis.org

13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769, 26089808579, 665043081119, 554688278429, 4090932431513069, 95405042230542329, 90616211958465842219, 810433818265726529159

Offset: 1

N. J. A. Sloane

nonn

The word "complete" indicates each chain is exactly n primes long (i.e., the chain cannot be a subchain of another one). Except for a(1), each term, by definition, is a Sophie Germain prime (A005384) as is each element except the last in each chain; each element after the first in each chain is a safe prime (A005385), so interior elements are both.

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

C. Caldwell, The Prime Number Glossary: Cunningham Chains.
G. Löh, Long chains of nearly doubled primes, Math. Comp. vol. 53 no. 188 (1989) pp 751-759.
W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains
Jonathan P. Sorenson, Jonathan Webster, Two Algorithms to Find Primes in Patterns, arXiv:1807.08777 [math.NT], 2018.

Cf. (A005384 and A005385), A007700, A023272, A023302, A023330, A057331, A005603.

a(13) found by Jack Brennen; a(14) found by Paul Jobling (Paul.Jobling(AT)WhiteCross.com) [Oct 23 2000]
Better description from Rick L. Shepherd, Jul 07 2004
a(15) found by Jonathan Webster and Jonathan Sorenson, added Jun 26 2018
a(16) found by Phil Carmody and Paul Jobling, Feb 2002, and added by Mauro Fiorentini, Feb 21 2025

A053176 Primes p such that 2p+1 is composite.

Original entry on oeis.org

7, 13, 17, 19, 31, 37, 43, 47, 59, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 127, 137, 139, 149, 151, 157, 163, 167, 181, 193, 197, 199, 211, 223, 227, 229, 241, 257, 263, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 347, 349, 353, 367, 373, 379, 383

Offset: 1

Enoch Haga, Feb 29 2000

Primes not in A005384 = non-Sophie Germain primes.
Also, numbers n such that odd part of A005277(n) is prime. Proof by John Renze, Sep 30 2004
Sequence gives primes p such that B(2p) has denominator 6, where B(2n) are the Bernoulli numbers. - Benoit Cloitre, Feb 06 2002
Sequence gives all n such that the equation phi(x)=2n has no solution. - Benoit Cloitre, Apr 07 2002
A010051(a(n))*(1-A156660(a(n))) = 1; subsequence of A138887. - Reinhard Zumkeller, Feb 18 2009
Mersenne prime exponents > 3 must be in the union of this sequence and (A002144). - Roderick MacPhee, Jan 12 2017

			17 is a term because 2*17 + 1 = 35 is composite.

T. D. Noe, Table of n, a(n) for n = 1..10000
Lambert A'Campo, Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation, arXiv:2006.06737 [math.NT], 2020.

Cf. A005384, A005385, A059452, A059453, A059454, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156543, A156542.

[p: p in PrimesUpTo(12200) | not IsPrime(2*p+1)]; // Vincenzo Librandi, Jun 18 2015

Select[Prime[Range[1000]], ! PrimeQ[2 # + 1] &] (* Vincenzo Librandi, Jun 18 2015 *)

list(lim)=select(p->!isprime(2*p+1),primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

a(n) ~ n log n. - Charles R Greathouse IV, Feb 20 2012

A059455 Safe primes which are also Sophie Germain primes.

Original entry on oeis.org

5, 11, 23, 83, 179, 359, 719, 1019, 1439, 2039, 2063, 2459, 2819, 2903, 2963, 3023, 3623, 3779, 3803, 3863, 4919, 5399, 5639, 6899, 6983, 7079, 7643, 7823, 10163, 10799, 10883, 11699, 12203, 12263, 12899, 14159, 14303, 14699, 15803, 17939

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Primes p such that both (p-1)/2 and 2*p+1 are prime.
Except for 5, all are congruent to 11 modulo 12.
Primes "inside" Cunningham chains of first kind.
Infinite under Dickson's conjecture. - Charles R Greathouse IV, Jul 18 2012
See A162019 for the subset of a(n) that are "reproduced" by the application of the transformations (a(n)-1)/2 and 2*a(n)+1 to the set a(n). - Richard R. Forberg, Mar 05 2015

			83 is a term because it is prime and 2*83+1 = 167 and (83-1)/2 = 41 are both primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A005385.

Cf. A053176, A059452, A059453, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660, A156877, A162019.

[p: p in PrimesUpTo(20000) |IsPrime((p-1) div 2) and IsPrime(2*p+1)]; // Vincenzo Librandi, Oct 31 2014

lst={}; Do[p=Prime[n]; If[PrimeQ[(p-1)/2]&&PrimeQ[2*p+1], AppendTo[lst, p]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 02 2008 *)
Select[Prime[Range[1000]], AllTrue[{(# - 1)/2, 2 # + 1}, PrimeQ] &] (* requires Mathematica 10+; Feras Awad, Dec 19 2018 *)

forprime(p=2,1e5,if(isprime(p\2)&&isprime(2*p+1),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

from itertools import count, islice
from sympy import isprime, prime
def A059455_gen(): # generator of terms
    return filter(lambda p:isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059455_list = list(islice(A059455_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156660(a(n))*A156659(a(n)) = 1; A156877 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

A023272 Primes that remain prime through 3 iterations of the function f(x) = 2*x + 1.

Original entry on oeis.org

2, 5, 89, 179, 359, 509, 1229, 1409, 2699, 3539, 6449, 10589, 11549, 11909, 12119, 17159, 19709, 19889, 22349, 26189, 27479, 30389, 43649, 53639, 53849, 55229, 57839, 60149, 61409, 63419, 66749, 71399, 74699, 75329, 82499, 87539, 98369, 101399, 104369

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+1, 4*p+3 and 8*p+7 are also primes. - Vincenzo Librandi, Aug 04 2010

For n > 2, a(n) == 29 (mod 30). - Zak Seidov, Jan 31 2013

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Intersection of A007700 and A023231.

Cf. A005384, A005385, A023302, A023330, A057331, A005602.

[n: n in [1..100000] | IsPrime(n) and IsPrime(2*n+1) and IsPrime(4*n+3) and IsPrime(8*n+7)] // Vincenzo Librandi, Aug 04 2010

p:=2: for n from 1 to 5000 do if(isprime(2*p+1) and isprime(4*p+3) and isprime(8*p+7))then printf("%d, ",p): fi: p:=nextprime(p): od: # Nathaniel Johnston, Jun 30 2011

Select[Prime[Range[10^3*4]], PrimeQ[a1=2*#+1] && PrimeQ[a2=2*a1+1] && PrimeQ[a3=2*a2+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Join[{2, 5}, Select[Range[89, 104369, 30], PrimeQ[#] && PrimeQ[2*# + 1] && PrimeQ[4*# + 3] && PrimeQ[8*# + 7] &]] (* Zak Seidov, Jan 31 2013 *)
p3iQ[n_]:=AllTrue[NestList[2#+1&,n,3],PrimeQ]; Join[{2,5},Select[ Range[ 89,200000,30],p3iQ]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 30 2019 *)

is(n)=isprime(n)&&isprime(2*n+1)&&isprime(4*n+3)&&isprime(8*n+7) \\ Charles R Greathouse IV, Mar 21 2013

A023330 Primes that remain prime through 5 iterations of function f(x) = 2x + 1.

Original entry on oeis.org

89, 63419, 127139, 405269, 810809, 1069199, 1122659, 1178609, 1333889, 1598699, 1806089, 1958249, 2164229, 2245319, 2329469, 2606069, 2848949, 3241289, 3339989, 3784199, 3962039, 4088879, 4328459, 4444829, 4658939, 4664249, 4894889, 4897709, 5132999

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+1, 4*p+3, 8*p+7, 16*p+15 and 32*p+31 are also primes. - Vincenzo Librandi, Aug 04 2010

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A007700, A023272, A023302, A057331, A005602.

[n: n in [1..5000000] | forall{2^i*n+2^i-1: i in [0..5] | IsPrime(2^i*n+2^i-1)}]; // Vincenzo Librandi, Aug 04 2010

Select[Prime[Range[10^5]], PrimeQ[a1=2*#+1] && PrimeQ[a2=2*a1+1] && PrimeQ[a3=2*a2+1] && PrimeQ[a4=2*a3+1] && PrimeQ[a5=2*a4+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

is(n)=isprime(n) && isprime(2*n+1) && isprime(4*n+3) && isprime(8*n+7) && isprime(16*n+15) && isprime(32*n+31) \\ Charles R Greathouse IV, Jul 01 2013

from sympy import prime, isprime
A023330_list = [p for p in (prime(n) for n in range(1,10**5)) if all([isprime(2**m*(p+1)-1) for m in range(1,6)])] # Chai Wah Wu, Sep 09 2014

a(n) == 29 (mod 30). - Zak Seidov, Jan 31 2013

A023302 Primes that remain prime through 4 iterations of function f(x) = 2x + 1.

Original entry on oeis.org

2, 89, 179, 53639, 53849, 61409, 63419, 66749, 126839, 127139, 143609, 167729, 186149, 206369, 254279, 268049, 296099, 340919, 405269, 422069, 446609, 539009, 594449, 607319, 658349, 671249, 725009, 775949, 810539, 810809, 812849, 819509

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+1, 4*p+3, 8*p+7 and 16*p+15 are also primes. - Vincenzo Librandi, Aug 04 2010

For n > 1, a(n) == 29 (mod 30). One should use it in codes. - Zak Seidov, Jan 31 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A007700, A023272, A023330, A057331, A005602.

[n: n in [1..1200000] | IsPrime(n) and IsPrime(2*n+1) and IsPrime(4*n+3) and IsPrime(8*n+7) and IsPrime(16*n+15)] // Vincenzo Librandi, Aug 04 2010

Select[Prime[Range[10^4*4]], PrimeQ[a1=2*#+1] && PrimeQ[a2=2*a1+1] && PrimeQ[a3=2*a2+1] && PrimeQ[a4=2*a3+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Join[{2},Select[Range[29,820000,30],And@@PrimeQ[NestList[2#+1&,#,4]]&]] (* Harvey P. Dale, Apr 03 2013 *)

is(n)=isprime(n) && isprime(2*n+1) && isprime(4*n+3) && isprime(8*n+7) && isprime(16*n+15) \\ Charles R Greathouse IV, Jul 01 2013

A309906 a(n) is the smallest number of divisors of p^n - 1 that may possibly occur for arbitrarily large primes p.

Original entry on oeis.org

4, 32, 8, 160, 8, 384, 8, 384, 16, 256, 8, 7680, 8, 128, 32, 1792, 8, 4096, 8, 3840, 32, 256, 8, 36864, 16, 128, 32, 2560, 8, 24576, 8, 4096, 32, 128, 32, 327680, 8, 128, 32, 36864, 8, 18432, 8, 2560, 128, 256, 8, 344064, 16, 1024, 32, 2560, 8, 20480, 32

Offset: 1

Jon E. Schoenfield, Aug 21 2019

nonn

The existence of infinitely many primes p such that p^n - 1 has exactly a(n) divisors is conjectured. E.g., although it is known that p-1 has fewer than 4 divisors for only finitely many primes p (see Example section), it is not known whether there exist infinitely many primes p such that p-1 has exactly 4 divisors. (Thanks to Jianing Song, who pointed out the need for this clarification.) - Jon E. Schoenfield, Mar 04 2021
For each prime q, every number k that has exactly q divisors is a prime power k = p^(q-1) for some prime p. As a result, a(q-1) can be useful in identifying numbers of the form p^(q-1) - 1 that are terms of A161460 (see Example section).
From Bernard Schott, Aug 22 2019: (Start)
For n prime >= 3, a(n) = 8;
for n = q^2, q prime >= 3, a(n) = 16. (End)

			a(1) = 4: The only primes p for which p-1 has fewer than 4 divisors are 2, 3, and 5; for all primes p > 5, p-1 has at least 4 divisors, and the terms in A005385 (Safe primes) except 5 are primes p such that p-1 has exactly 4 divisors.
a(2) = 32: p^2 - 1 = (p-1)*(p+1) has fewer than 32 divisors only for p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 47, and 73; for all primes p such that the product of the 3-smooth parts of p-1 and p+1 is 24 and p-1 and p+1 each have one prime factor > 3, p^2 - 1 has exactly 32 divisors (see A341658).
a(4) = 160: primes p such that p^4 - 1 has exactly 160 divisors are plentiful (see A341662), but only p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 59, 61, 71, 79, and 101 yield tau(p^4 - 1) < 160. Of these, p = 13, 29, 59, and 61 all give tau(p^4 - 1) = 80; 37 and 101 both give 120 divisors; and 41 and 71 both give 144. For each of the ten remaining primes (p = 2, 3, 5, 7, 11, 17, 19, 23, 31, 79), the value of tau(p^4 - 1) is unique, so each of those ten values of p^4 - 1 is a term in A161460.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A005385, A006863, A079612, A161460, A341658, A341662, A341670.

f(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)); ); ); res; ); } \\ A079612
a(n) = numdiv(f(n))*2^numdiv(n); \\ Michel Marcus, Aug 22 2019

a(n) = A000005(A079612(n))*2^A000005(n).

a(n) = 2^(A000005(n)+1) for odd n. - Jianing Song, Dec 05 2021

Name edited by Jon E. Schoenfield, Mar 04 2021

A072055 a(n) = 2*prime(n)+1.

Original entry on oeis.org

5, 7, 11, 15, 23, 27, 35, 39, 47, 59, 63, 75, 83, 87, 95, 107, 119, 123, 135, 143, 147, 159, 167, 179, 195, 203, 207, 215, 219, 227, 255, 263, 275, 279, 299, 303, 315, 327, 335, 347, 359, 363, 383, 387, 395, 399, 423, 447, 455, 459, 467, 479

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

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

Cf. A000040, A005385, A023589, A023590, A023591, A023592, A023593, A072059, A023595, A072060, A072056, A072057, A072058, A072192, A165286, A278230.
One less than A089241. After the initial term equal to A166496.
Row 4 of A286625, column 4 of A286623.

a072055 = (+ 1) . (* 2) . a000040  -- Reinhard Zumkeller, Oct 10 2013

2*Prime[Range[60]]+1  (* Harvey P. Dale, Mar 31 2011 *)

a(n) = A089241(n)-1.

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

cp's OEIS Frontend

A007700 Numbers n such that n, 2n+1, and 4n+3 all prime.

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A005602 Smallest prime beginning a complete Cunningham chain of length n (of the first kind).

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A053176 Primes p such that 2p+1 is composite.

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A059455 Safe primes which are also Sophie Germain primes.

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A023272 Primes that remain prime through 3 iterations of the function f(x) = 2*x + 1.

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A023330 Primes that remain prime through 5 iterations of function f(x) = 2x + 1.

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A023302 Primes that remain prime through 4 iterations of function f(x) = 2x + 1.

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