A007528 -id:A007528 - cp's OEIS frontend

A005384 Sophie Germain primes p: 2p+1 is also prime.

2, 3, 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, 1559

Offset: 1

N. J. A. Sloane

Then 2p+1 is called a safe prime: see A005385.
Primes p such that the equation phi(x) = 2p has solutions, where phi is the totient function. See A087634 for another such collection of primes. - T. D. Noe, Oct 24 2003
Subsequence of A117360. - Reinhard Zumkeller, Mar 10 2006
Let q = 2n+1. For these n (and q), the difference of two cyclotomic polynomials can be written as a cyclotomic polynomial in x^2: Phi(q,x) - Phi(2q,x) = 2x Phi(n,x^2). - T. D. Noe, Jan 04 2008
A Sophie Germain prime p is 2, 3 or of the form 6k-1, k >= 1, i.e., p = 5 (mod 6). A prime p of the form 6k+1, k >= 1, i.e., p = 1 (mod 6), cannot be a Sophie Germain prime since 2p+1 is divisible by 3. - Daniel Forgues, Jul 31 2009
Also solutions to the equation: floor(4/A000005(2*n^2+n)) = 1. - Enrique Pérez Herrero, May 03 2012
In the spirit of the conjecture related to A217788, we conjecture that for any integers n >= m > 0 there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) is prime. - Zhi-Wei Sun, Mar 26 2013
If k is the product of a Sophie Germain prime p and its corresponding safe prime 2p+1, then a(n) = (k-phi(k))/3, where phi is Euler's totient function. - Wesley Ivan Hurt, Oct 03 2013
Giovanni Resta found the first Sophie Germain prime which is also a Brazilian number (A125134), 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73. - _Bernard Schott, Mar 07 2019
For all Sophie Germain primes p >= 5, 2*p + 1 = min(A, B) where A is the smallest prime factor of 2^p - 1 and B the smallest prime factor of (2^p + 1) / 3. - Alain Rocchelli, Feb 01 2023
Consider a pair of numbers (p, 2*p+1), with p >= 3. Then p is a Sophie Germain prime iff (p-1)!^2 + 6*p == 1 (mod p*(2*p+1)). - Davide Rotondo, May 02 2024

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Peretti, The quantity of Sophie Germain primes less than x, Bull. Number Theory Related Topics, Vol. 11, No. 1-3 (1987), pp. 81-92.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 76, 227-230.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

J. S. Cheema, Table of n, a(n) for n = 1..100000. [This replaces an earlier b-file computed by T. D. Noe]
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
P. Bruillard, S.-H. Ng, E. Rowell and Z. Wang, On modular categories, arXiv preprint arXiv:1310.7050 [math.QA], 2013.
Chris K. Caldwell, The Prime Glossary, Sophie Germain Prime
Andrea Del Centina, Letters of Sophie Germain preserved in Florence Historia Mathematica, Vol. 32 (2005), pp. 60-75.
Harvey Dubner, Large Sophie Germain Primes, Math. Comp., Vol. 65, No. 213 (1996), pp. 393-396.
Luis H. Gallardo and Olivier Rahavandrainy, There are finitely many even perfect polynomials over F_p with p+1 irreducible divisors, Acta Mathematica Universitatis Comenianae, Vol. 83, No. 2, 2016, 261-275.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Reikku Kulon, Sublinear arbitrary precision generator of Sophie Germain and safe primes in C (public domain).
Henri Lifchitz, A new and simpler primality test for Sophie-Germain numbers(q=2*p+1).
Victor Meally, Letter to N. J. A. Sloane, no date.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
Frans Oort, Prime numbers, 2013.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 31, 114.
Larry Riddle, Sophie Germain and Fermat's Last Theorem, Agnes Scott College, Math. Dept., Jul, 1999.
Carlos Rivera, Puzzle 1122. OEIS A005385, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 1140. Test for Sophie Germain primes, The Prime Puzzles & Problems Connection.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Anthony G. Shannon, Peter J.-S. Shiue, Tian-Xiao He, and Christopher Saito, On the special cases of Carmichael's totient conjecture, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 504-534.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.
Agoh Takashi, On Sophie Germain primes, Number theory (Liptovský Ján, 1999), Tatra Mt. Math. Publ., Vol. 20 (2000), pp. 65-73.
Terence Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Vmoraru, PlanetMath.org, Sophie Germain prime.
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.
Eric Weisstein's World of Mathematics, Sophie Germain Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Sophie Germain prime.
Samuel Yates, Sophie Germain primes, in "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991, pp. 882-886.

Cf. A005385, A057331, A005602, A087634.
Cf. also A000355, A156541, A156542, A156592, A161896, A156660, A156874, A092816, A023212, A007528 (primes of the form 6k-1).
For primes p that remains prime through k iterations of the function f(x) = 2x + 1: this sequence (k=1), A007700 (k=2), A023272 (k=3), A023302 (k=4), A023330 (k=5), A278932 (k=6), A138025 (k=7), A138030 (k=8).

Filtered([1..1600],p->IsPrime(p) and IsPrime(2*p+1)); # Muniru A Asiru, Mar 06 2019

[ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; // Klaus Brockhaus, Jan 01 2009

A:={}: for n from 1 to 246 do if isprime(2*ithprime(n)+1) then A:=A union {ithprime(n)} fi od: A:=A; # Emeric Deutsch, Dec 09 2004

Select[Prime[Range[1000]],PrimeQ[2#+1]&]
lst = {}; Do[If[PrimeQ[n + 1] && PrimeOmega[n] == 2, AppendTo[lst, n/2]], {n, 2, 10^4}]; lst (* Hilko Koning, Aug 17 2021 *)

select(p->isprime(2*p+1), primes(1000)) \\ In old PARI versions <= 2.4.2, use select(primes(1000), p->isprime(2*p+1)).

forprime(n=2, 10^3, if(ispseudoprime(2*n+1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

is_A005384=(p->isprime(2*p+1)&&isprime(p));
  {A005384_vec(N=100,p=1)=vector(N,i,until(isprime(2*p+1),p=nextprime(p+1));p)} \\ M. F. Hasler, Mar 03 2020

from sympy import isprime, nextprime
def ok(p): return isprime(2*p+1)
def aupto(limit): # only test primes
  alst, p = [], 2
  while p <= limit:
    if ok(p): alst.append(p)
    p = nextprime(p)
  return alst
print(aupto(1559)) # Michael S. Branicky, Feb 03 2021

a(n) mod 10 <> 7. - Reinhard Zumkeller, Feb 12 2009
A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= n. - Reinhard Zumkeller, Feb 18 2009
tau(4*a(n) + 2) = tau(4*a(n)) - 2, for n > 1. - Arkadiusz Wesolowski, Aug 25 2012
eulerphi(4*a(n) + 2) = eulerphi(4*a(n)) + 2, for n > 1. - Arkadiusz Wesolowski, Aug 26 2012
A005097 INTERSECT A000040. - R. J. Mathar, Mar 23 2017
Sum_{n>=1} 1/a(n) is in the interval (1.533944198, 1.8026367) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021
a(n) >> n log^2 n. - Charles R Greathouse IV, Jul 25 2024

A002476 Primes of the form 6m + 1.

Original entry on oeis.org

7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619

Offset: 1

N. J. A. Sloane

Equivalently, primes of the form 3m + 1.
Rational primes that decompose in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p dividing Sum_{k=0..p} binomial(2k, k) - 3 = A006134(p) - 3. - Benoit Cloitre, Feb 08 2003
Primes p such that tau(p) == 2 (mod 3) where tau(x) is the Ramanujan tau function (cf. A000594). - Benoit Cloitre, May 04 2003
Primes of the form x^2 + xy - 2y^2 = (x+2y)(x-y). - N. J. A. Sloane, May 31 2014
Primes of the form x^2 - xy + 7y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that p^2 divides Sum_{m=1..2(p-1)} Sum_{k=1..m} (2k)!/(k!)^2. - Alexander Adamchuk, Jul 04 2006
A006512 larger than 5 (Greater of twin primes) is a subsequence of this. - Jonathan Vos Post, Sep 03 2006
A039701(A049084(a(n))) = A134323(A049084(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2007
Also primes p such that the arithmetic mean of divisors of p^2 is an integer: sigma_1(p^2)/sigma_0(p^2) = C. (A000203(p^2)/A000005(p^2) = C). - Ctibor O. Zizka, Sep 15 2008
Fermat knew that these numbers can also be expressed as x^2 + 3y^2 and are therefore not prime in Z[omega], where omega is a complex cubic root of unity. - Alonso del Arte, Dec 07 2012
Primes of the form x^2 + xy + y^2 with x < y and nonnegative. Also see A007645 which also applies when x=y, adding an initial 3. - Richard R. Forberg, Apr 11 2016
For any term p in this sequence, let k = (p^2 - 1)/6; then A016921(k) = p^2. - Sergey Pavlov, Dec 16 2016; corrected Dec 18 2016
For the decomposition p=x^2+3*y^2, x(n) = A001479(n+1) and y(n) = A001480(n+1). - R. J. Mathar, Apr 16 2024

			Since 6 * 1 + 1 = 7 and 7 is prime, 7 is in the sequence. (Also 7 = 2^2 + 3 * 1^2 = (2 + sqrt(-3))(2 - sqrt(-3)).)
Since 6 * 2 + 1 = 13 and 13 is prime, 13 is in the sequence.
17 is prime but it is of the form 6m - 1 rather than 6m + 1, and is therefore not in the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
David A. Cox, Primes of the Form x^2 + ny^2. New York: Wiley (1989): 8.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 261.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. Banderier, Calcul de (-3/p)
Barry Brent, Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups, Integers (2024) Vol. 24, Art. No. A18. See p. 13.
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.
K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.
Neville Robbins, On the Infinitude of Primes of the Form 3k+1, Fib. Q., 43,1 (2005), 29-30.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Index to sequences related to decomposition of primes in quadratic fields

Cf. A045331, A242660.
For values of m see A024899. Primes of form 3n - 1 give A003627.
These are the primes arising in A024892, A024899, A034936.
A091178 gives prime index.
Cf. A006512, A007528.
Subsequence of A016921 and of A050931.
Cf. A004611 (multiplicative closure).

Filtered(List([0..110],k->6*k+1),n-> IsPrime(n)); # Muniru A Asiru, Mar 11 2019

a002476 n = a002476_list !! (n-1)
a002476_list = filter ((== 1) . (`mod` 6)) a000040_list
-- Reinhard Zumkeller, Jan 15 2013

(#~ 1&p:) >: 6 * i.1000 NB. Stephen Makdisi, May 01 2018

[n: n in [1..700 by 6] | IsPrime(n)]; // Vincenzo Librandi, Apr 05 2011

a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];

Select[6*Range[100] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 06 2006 *)

select(p->p%3==1,primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n >= 1} 1/a(n)^2 = A175644.
Sum_{n >= 1} 1/a(n)^3 = A175645. (End)
a(n) = 6*A024899(n) + 1. - Zak Seidov, Aug 31 2016
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 1/A175646.
Product_{k>=1} (1 + 1/a(k)^2) = A334481.
Product_{k>=1} (1 - 1/a(k)^3) = A334478.
Product_{k>=1} (1 + 1/a(k)^3) = A334477. (End)
Legendre symbol (-3, a(n)) = +1 and (-3, A007528(n)) = -1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A003627 Primes of the form 3n-1.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane and Mira Bernstein

Inert rational primes in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p such that 1+x+x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1. - Benoit Cloitre, Feb 08 2003
A039701(A049084(a(n))) = 2; A134323(A049084(a(n))) = -1. - Reinhard Zumkeller, Oct 21 2007
The set of primes of the form 3n - 1 is a superset of the set of lesser of twin primes larger than three (A001359). - Paul Muljadi, Jun 05 2008
Primes of this form do not occur in or as divisors of {n^2+n+1}. See A002383 (n^2+n+1 = prime), A162471 (prime divisors of n^2+n+1 not in A002383), and A002061 (numbers of the form n^2-n+1). - Daniel Tisdale, Jul 04 2009
Or, primes not in A007645. A003627 UNION A007645 = A000040. Also, primes of the form 6*k-5/2-+3/2. - Juri-Stepan Gerasimov, Jan 28 2010
Except for first term "2", all these prime numbers are of the form: 6*n-1. - Vladimir Joseph Stephan Orlovsky, Jul 13 2011
A088534(a(n)) = 0. - Reinhard Zumkeller, Oct 30 2011
For n>1: Numbers k such that (k-4)! mod k =(-1)^(floor(k/3)+1)*floor((k+1)/6), k>4. - Gary Detlefs, Jan 02 2012
Binomial(a(n),3)/a(n)= (3*A024893(n)^2+A024893(n))/2, n>1. - Gary Detlefs, May 06 2012
For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - Michel Marcus, Nov 12 2017
2 adjoined to A007528. - David A. Corneth, Nov 12 2017
For n >= 2 there exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
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
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Eisenstein Prime
Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand., Vol 35 (1974), 309-317.
Index to sequences related to decomposition of primes in quadratic fields

Primes of form 3n+1 give A002476.
These are the primes arising in A024893, A087370, A088879. A091177 gives prime index.
Cf. A001359, A007528, A007645, A221717, A057145.
Subsequence of A034020.

a003627 n = a003627_list !! (n-1)
a003627_list = filter ((== 2) . (`mod` 3)) a000040_list
-- Reinhard Zumkeller, Oct 30 2011

[n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 2011

t1 := {}; for n from 0 to 500 do if isprime(3*n+2) then t1 := {op(t1),3*n+2}; fi; od: A003627 := convert(t1,list);

Select[Range[-1, 600, 3], PrimeQ[#] &] (* Vincenzo Librandi, Jun 17 2015 *)
Select[Prime[Range[200]],Mod[#,3]==2&] (* Harvey P. Dale, Jan 31 2023 *)

is(n)=n%3==2 && isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n>=1} 1/a(n)^2 = 0.30792... = A085548 - 1/9 - A175644.
Sum_{n>=1} 1/a(n)^3 = 0.134125... = A085541 - 1/27 - A175645. (End)

A075677 Reduced Collatz function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (3k+1)/2^r, with r as large as possible.

Original entry on oeis.org

1, 5, 1, 11, 7, 17, 5, 23, 13, 29, 1, 35, 19, 41, 11, 47, 25, 53, 7, 59, 31, 65, 17, 71, 37, 77, 5, 83, 43, 89, 23, 95, 49, 101, 13, 107, 55, 113, 29, 119, 61, 125, 1, 131, 67, 137, 35, 143, 73, 149, 19, 155, 79, 161, 41, 167, 85, 173, 11, 179, 91, 185, 47, 191, 97, 197

Offset: 1

T. D. Noe, Sep 25 2002

The even-indexed terms a(2i+2) = 6i+5 = A016969(i), i >= 0 [Comment corrected by Bob Selcoe, Apr 06 2015]. The odd-indexed terms are the same as A067745. Note that this sequence is A016789 with all factors of 2 removed from each term. Also note that a(4i-1) = a(i). No multiple of 3 is in this sequence. See A075680 for the number of iterations of R required to yield 1.
From Bob Selcoe, Apr 06 2015: (Start)
All numbers in this sequence appear infinitely often.
From Eq. 1 and Eq. 2 in Formulas: Eq. 1 is used with 1/3 of the numbers in this sequence, Eq. 2 is used with 2/3 of the numbers.
(End)
Empirical: For arbitrary m, Sum_{n=2..A007583(m)} (a(n) - a(n-1)) = 0. - Fred Daniel Kline, Nov 23 2015
From Wolfdieter Lang, Dec 07 2021: (Start)
Only positive numbers congruent to 1 or 5 modulo 6 appear.
i) For the sequence entry with value A016921(m), for m >= 0, that is, a value from {1, 7, 13, ...}, the indices n are given by the row of array A178415(2*m+1, k), for k >= 1.
ii) For the sequence entry with value A007528(m), for m >= 1, that is, a value from {5, 11, 17, ...}, the indices n are given by the row of array A178415(2*m, k), for k >= 1.
See also the array A347834 with permuted row numbers and columns k >= 0. (End)

			a(11) = 1 because 21 is the 11th odd number and R(21) = 64/64 = 1.
From _Wolfdieter Lang_, Dec 07 2021: (Start)
i) 1 (mod 6) entry 1 = A016921(0) appears for n = A178415(1, k) = A347834(1, k-1) (the arrays), for k >= 1, that is, for {1, 5, 21, ..} = A002450.
ii) 5 (mod 6) entry 11 = A007528(2) appears for n = A178415(4, k) = A347835(3, k-1) (the arrays), for k >= 1, that is, for {7, 29, 117, ..} = A072261. (End)

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E16, pp. 330-336.
Victor Klee and Stan Wagon, Old and new unsolved problems in plane geometry and number theory, The Mathematical Association of America, 1991, p. 225, C(2n+1) = a(n+1), n >= 0.
Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see p. 57, also (90-9), p. 306.

T. D. Noe, Table of n, a(n) for n = 1..1000
Marc Chamberland, Una actualizacio del problema 3x + 1, Butl. Soc. Catalana Mat. (18), 19-45, 2003.
Jeffrey C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.
Eric Weisstein's World of Mathematics, Collatz Problem.
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A000265, A000302, A002450, A007528, A016789, A016921, A016969, A065677, A072197, A072261, A075680, A081294, A178415, A191669, A329480, A347834.
Cf. A006370, A014682 (for non-reduced Collatz maps), A087230 (A371093), A371094.
Odd bisection of A139391.
Even bisection of A067745, which is also the odd bisection of this sequence.
After the initial 1, the second leftmost column of A256598.
Row 2 of A372283.

a075677 = a000265 . subtract 2 . (* 6) -- Reinhard Zumkeller, Jan 08 2014

f:=proc(n) local t1;
if n=1 then RETURN(1) else
t1:=3*n+1;
while t1 mod 2 = 0 do t1:=t1/2; od;
RETURN(t1); fi;
end;
# N. J. A. Sloane, Jan 21 2011

nextOddK[n_] := Module[{m=3n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]
v[x_] := IntegerExponent[x, 2]; f[x_] := (3*x + 1)/2^v[3*x + 1]; Table[f[2*n - 1], {n, 66}] (* L. Edson Jeffery, May 06 2015 *)

a(n)=n+=2*n-1;n>>valuation(n,2) \\ Charles R Greathouse IV, Jul 05 2013

from sympy import divisors
def a(n):
    return max(d for d in divisors(n) if d % 2)
print([a(6*n - 2) for n in range(1, 101)]) # Indranil Ghosh, Apr 15 2017, after formula by Reinhard Zumkeller

a(n) = A000265(6*n-2) = A000265(3*n-1). - Reinhard Zumkeller, Jan 08 2014
From Bob Selcoe, Apr 05 2015: (Start)
For all n>=1 and for every k, there exists j>=0 dependent upon n and k such that either:
Eq. 1: a(n) = (3n-1)/2^(2j+1) when k = ((4^(j+1)-1)/3) mod 2^(2j+3). Alternatively: a(n) = A016789(n-1)/A081294(j+1) when k = A002450(j+1) mod A081294(j+2). Example: n=51; k=101 == 5 mod 32, j=1. a(51) = 152/8 = 19.
  or
Eq. 2: a(n) = (3n-1)/4^j when k = (5*2^(2j+1) - 1)/3 mod 4^(j+1). Alternatively: a(n) = A016789(n-1)/A000302(j) when k = A072197(j) mod A000302(j+1). Example: n=91; k=181 == 53 mod 64, j=2. a(91) = 272/16 = 17.
(End) [Definition corrected by William S. Hilton, Jul 29 2017]
a(n) = a(n + g*2^r) - 6*g, n > -g*2^r. Examples: n=59; a(59)=11, r=5. g=-1: 11 = a(27) = 5 - (-1)*6; g=1: 11 = a(91) = 17 - 1*6; g=2: 11 = a(123) = 23 - 2*6; g=3: 11 = a(155) = 29 - 3*6; etc. - Bob Selcoe, Apr 06 2015
a(n) = a((1 + (3*n - 1)*4^(k-1))/3), k>=1 (cf. A191669). - L. Edson Jeffery, Oct 05 2015
a(n) = a(4n-1). - Bob Selcoe, Aug 03 2017
a(n) = A139391(2n-1). - Antti Karttunen, May 06 2024
Sum_{k=1..n} a(k) ~ n^2. - Amiram Eldar, Aug 26 2024
G.f.: Sum_{k>=1} ((3 + 2*(-1)^k)*x^(3*2^(k - 1) - (-2)^k/3 + 1/3) + (3 - 2*(-1)^k)*x^(2^(k - 1) - (-2)^k/3 + 1/3))/(x^(2^k) - 1)^2. - Miles Wilson, Oct 26 2024

A039949 Primes of the form 30n - 13.

Original entry on oeis.org

17, 47, 107, 137, 167, 197, 227, 257, 317, 347, 467, 557, 587, 617, 647, 677, 797, 827, 857, 887, 947, 977, 1097, 1187, 1217, 1277, 1307, 1367, 1427, 1487, 1607, 1637, 1667, 1697, 1787, 1847, 1877, 1907, 1997, 2027, 2087, 2207, 2237, 2267, 2297, 2357, 2417

Offset: 1

Felice Russo

This linear form produces the most primes for n between 1 and 1000 (411/1000).
Primes congruent to 17 (mod 30). - Omar E. Pol, Aug 15 2007
Primes ending in 7 with (SOD-1)/3 non-integer where SOD is sum of digits. - Ki Punches
Or primes p such that (p mod 3) = (p mod 5) and (p mod 2) <> (p mod 3), (p > 2). - Mikk Heidemaa, Jan 19 2016

C. Clawson, Mathematical Mysteries, Plenum Press, 1996, p. 173

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

Cf. A132230-A132236, A141860.

[p: p in PrimesUpTo(3000) | p mod 30 in [17]]; // Vincenzo Librandi, Aug 04 2012

Select[Prime[Range[1000]],MemberQ[{17},Mod[#,30]]&] (* Vincenzo Librandi, Aug 04 2012 *)
Select[Range[17,3000,30],PrimeQ] (* Zak Seidov, Apr 15 2015 *)

select(n->n%30==17, primes(500)) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A158648(n)*30 + 17. - Ray Chandler, Apr 07 2009
Intersection of A030432 and A007528. - Ray Chandler, Apr 07 2009
a(n) = A141860(n+1). - Zak Seidov, Apr 15 2015

Extended by Ray Chandler, Apr 07 2009

A175646 Decimal expansion of the Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2).

Original entry on oeis.org

1, 0, 3, 4, 0, 1, 4, 8, 7, 5, 4, 1, 4, 3, 4, 1, 8, 8, 0, 5, 3, 9, 0, 3, 0, 6, 4, 4, 4, 1, 3, 0, 4, 7, 6, 2, 8, 5, 7, 8, 9, 6, 5, 4, 2, 8, 4, 8, 9, 0, 9, 9, 8, 8, 6, 4, 1, 6, 8, 2, 5, 0, 3, 8, 4, 2, 1, 2, 2, 2, 2, 4, 5, 8, 7, 1, 0, 9, 6, 3, 5, 8, 0, 4, 9, 6, 2, 1, 7, 0, 7, 9, 8, 2, 6, 2, 0, 5, 9, 6, 2, 8, 9, 9, 7

Offset: 1

R. J. Mathar, Aug 01 2010

The Euler product of the Riemann zeta function at 2 restricted to primes in A002476, which is the inverse of the infinite product (1-1/7^2)*(1-1/13^2)*(1-1/19^2)*...
There is a complementary Product_{primes p == 2 (mod 3)} 1/(1-1/p^2) = A333240 = 1.4140643908921476375655018190798... such that (this constant here)*1.4140643.../(1-1/3^2) = zeta(2) = A013661.
Because 1/(1-p^(-2)) = 1+1/(p^2-1), the complementary 1.414064... also equals Product_{primes p == 2 (mod 3)} (1+1/(p^2-1)), which appears in Eq. (1.8) of [Dence and Pomerance]. - R. J. Mathar, Jan 31 2013

			1.03401487541434188053903064441304762857896...

Peter Luschny, Table of n, a(n) for n = 1..1000 (terms 1..105 from Vaclav Kotesovec).
Thomas Dence and Carl Pomerance, Euler's Function in Residue Classes, Raman. J., Vol. 2 (1998) pp. 7-20, alternative link.
S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26.

Cf. A002476, A004611, A007528, A175647, A301429, A333240, A334478, A334480.

z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n) / 3:
evalf(4*Pi^2 / (27*mul(x(n), n=1..8)), 106); # Peter Luschny, Jan 17 2021

digits = 105;
precision = digits + 5;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
pA = Pi^2/9/pB ;
RealDigits[pA, 10, digits][[1]]
(* Jean-François Alcover, Jan 11 2021, after PARI code due to Artur Jasinski *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[3,1,2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)
z[n_] := Zeta[n] / Im[PolyLog[n, (-1)^(2/3)]];
x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n) / 3;
N[4 Pi^2 / (27 Product[x[n], {n, 8}]), 106] (* Peter Luschny, Jan 17 2021 *)

Equals 2*Pi^2 / (3^(7/2) * A301429^2). - Vaclav Kotesovec, May 12 2020

Equals Sum_{k>=1} 1/A004611(k)^2. - Amiram Eldar, Sep 27 2020

More digits from Vaclav Kotesovec, May 12 2020 and Jun 27 2020

A066796 a(n) = Sum_{i=1..n} binomial(2*i,i).

Original entry on oeis.org

2, 8, 28, 98, 350, 1274, 4706, 17576, 66196, 250952, 956384, 3660540, 14061140, 54177740, 209295260, 810375650, 3143981870, 12219117170, 47564380970, 185410909790, 723668784230, 2827767747950, 11061198475550, 43308802158650

Offset: 1

Benoit Cloitre, Jan 18 2002

nonn

Comments from Alexander Adamchuk, Jul 02 2006: (Start)
Every a(n) is divisible by prime 2, a(n)/2 = A079309(n).
a(n) is divisible by prime 3 only for n=12,30,36,84,90,108,120,... A083096.
a(p) is divisible by p^2 for primes p=5,11,17,23,29,41,47,... Primes of form 6n-1. A007528.
a(p-1) is divisible by p^2 for primes p=7,13,19,31,37,43,... Primes of form 6n+1. A002476.
Every a(n) from a((p-1)/2) to a(p-1) is divisible by prime p for p=7,13,19,31,37,43,... Primes of form 6n+1. A002476.
Every a(n) from a((p^2-1)/2) to a(p^2-1) is divisible by prime p>3.
a(p^2-1), a(p^2-2) and a(p^2-3) are divisible by p^2 for prime p>3.
a(p^2-4) is divisible by p^2 for prime p>5.
a(p^2-5) is divisible by p^2 for prime p>7.
a(p^2-6) is divisible by p^2 for prime p>7.
a(p^2-7) is divisible by p^2 for prime p>11.
a(p^2-8) is divisible by p^2 for prime p>13.
a(p^3) is divisible by p^2 for prime 2 and prime p=5,11,... Primes of form 6n-1. A007528.
a(p^3-1) is divisible by p^2 for prime p=7,13,... Primes of form 6n+1. A002476.
a(p^4-1) is divisible by p^2 for prime p>3. (End)
Mod[ a(3^k), 9 ] = 1 for integer k>0. Smallest number k such that 2^n divides a(k) is k(n) = {1,2,2,11,11,46,46,707,707,707,...}. Smallest number k such that 3^n divides a(k) is k(n) = {12,822,2466,...}. a(2(p-1)/3) is divisible by p^2 for prime p = {7,13,19,31,37,43,61,...} = A002476 Primes of form 6n+1. Every a(n) from a(p^2-(p+1)/2) to a(p^2-1) is divisible by p^2 for prime p>3. Every a(n) from a((4p+3)(p-1)/6) to a((2p+3)(p-1)/3) is divisible by p^2 for prime p = {7,13,19,31,37,43,61,...} = A002476 Primes of form 6n+1. - Alexander Adamchuk, Jan 04 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000 (Terms 1 to 200 computed by Harry J. Smith; terms 201 to 1000 computed by G. C. Greubel, Jan 15 2017)
Guo-Shuai Mao, Proof of a conjecture of Adamchuk, arXiv:2003.09810 [math.NT], 2020.
Guo-Shuai Mao, On a supercongruence conjecture of Z.-W. Sun, arXiv:2003.14221 [math.NT], 2020.
Guo-Shuai Mao, On some supercongruence conjectures of Z.-W. Sun, Nanjing Univ. Info. Sci. Tech. (China, 2023).
Guo-Shuai Mao, Proof of some congruences via the hypergeometric identities, Nanjing Univ. Info. Sci. Tech. (China, 2023).
Guo-Shuai Mao, On two pairs of congruence conjectures of Z.-W. Sun, Nanjing Univ. Sci. Tech. (China), ResearchGate, 2024.
Guo-Shuai Mao and Roberto Tauraso, Three pairs of congruences concerning sums of central binomial coefficients, arXiv:2004.09155 [math.NT], 2020.
Guo-Shuai Mao and Dong-Hui Zhang, On some congruences involving harmonic numbers and Bernoulli polynomials, ResearchGate, 2023. See pp. 1-2, 12.
Z.-W. Sun, Fibonacci numbers modulo cubes of primes, arXiv:0911.3060 [math.NT], 2009-2013; Taiwanese J. Math., to appear 2013. - From _N. J. A. Sloane_, Mar 01 2013
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Eric Weisstein's World of Mathematics, Binomial Sums.

Essentially the same as A079309 and A054114.
Equals A006134 - 1.
Cf. A002476, A006134, A007528, A079309, A083096.

Table[Sum[(2k)!/(k!)^2,{k,1,n}],{n,1,50}] (* Alexander Adamchuk, Jul 02 2006 *)
Table[Sum[Binomial[2k,k],{k,1,n}],{n,1,30}] (* Alexander Adamchuk, Jan 04 2007 *)

{ a=0; for (n=1, 200, write("b066796.txt", n, " ", a+=binomial(2*n, n)) ) } \\ Harry J. Smith, Mar 27 2010

a(n) = sum(i=1, n, binomial(2*i,i)); \\ Michel Marcus, Jan 04 2016

a(n) = A006134(n) - 1; generating function: (sqrt(1-4*x)-1)/(sqrt(1-4*x)*(x-1)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 11 2003, corrected by Vaclav Kotesovec, Nov 06 2012
a(n) = Sum_{k=1..n}(2k)!/(k!)^2. - Alexander Adamchuk, Jul 02 2006
a(n) = Sum_{k=1..n}binomial(2k,k). - Alexander Adamchuk, Jan 04 2007
a(n) ~ 2^(2*n+2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 06 2012

A024898 Positive integers k such that 6*k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 15, 17, 18, 19, 22, 23, 25, 28, 29, 30, 32, 33, 38, 39, 40, 42, 43, 44, 45, 47, 49, 52, 53, 58, 59, 60, 64, 65, 67, 70, 72, 74, 75, 77, 78, 80, 82, 84, 85, 87, 93, 94, 95, 98, 99, 100, 103, 107, 108, 109, 110, 113, 114, 117, 120, 124, 127, 129, 133

Offset: 1

Clark Kimberling

nonn

Actual primes are A007528.

Number of terms less than 10^r, r=1,2,3,...: 8, 56, 397, 3040, 24571, 206502, 1781237, ... - Muniru A Asiru, Jan 26 2018

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 3000 terms from Zak Seidov)

Cf. A007528, A046953 (complement).

Filtered([1..1000], k->IsPrime(6*k-1)); # Muniru A Asiru, Jan 26 2018

[n: n in [0..1000]| IsPrime(6*n-1)]; // Vincenzo Librandi, Nov 20 2010

select(k -> isprime(6*k-1), [$1..10^3]); # Muniru A Asiru, Jan 26 2018

Select[Range[150], PrimeQ[6 # - 1] &] (* Bruno Berselli, Jul 14 2014 *)

A345992 Let m = A344005(n) = smallest m such that n divides m*(m+1); a(n) = gcd(n,m).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 5, 1, 1, 2, 1, 4, 3, 2, 1, 8, 1, 2, 1, 7, 1, 5, 1, 1, 11, 2, 7, 4, 1, 2, 3, 5, 1, 6, 1, 11, 9, 2, 1, 3, 1, 2, 17, 4, 1, 2, 5, 7, 3, 2, 1, 15, 1, 2, 9, 1, 5, 11, 1, 4, 23, 14, 1, 8, 1, 2, 3, 19, 7, 6, 1, 5, 1, 2, 1, 4, 17, 2, 29, 8, 1, 9, 13, 23, 3, 2, 19, 32, 1, 2, 11, 4, 1, 17

Offset: 1

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 15 2021

nonn

By definition, a(n)*A345993(n) = n.
a(n) is even iff n/2 is in A344000. This is true, but essentially trivial, and does not provide any insight into either sequence.
Empirical: For n >= 3, a(n) <= n/3, and a(n) = n/3 iff n is in 3*{2^odd, primes == -1 mod 6}.
If n = 2*p^k where p is an odd prime then m = A344005(n) = p^k - 1 and a(n) = 2. Conversely, it appears that if a(n) = 2 then n is twice an odd prime power. (Corrected by Antti Karttunen, Jun 14 2022)
a(n) = 1 if n is a prime power. - Chai Wah Wu, Jun 01 2022
From Antti Karttunen, Jun 14 2022: (Start)
Conversely, if a(n) = 1 [i.e., A345993(n) = n] then n is a power of prime. (This follows from N. J. A. Sloane's Jul 11 2021 theorem given in A344005).
Apparently, a(n) = 3 iff n = A354984(k) = 3*A137827(k), for some k >= 1.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A011772, A137827, A182665, A344000, A344005, A345993, A345994, A345995, A354930, A354931 (the least occurrence of each n=1..), A354984.

Cf. also A007528, A051119, A284600.

# load Findm from A344005
ans:=[];
for n from 1 to 40 do t1:=Findm(n)[1]; ans:=[op(ans), igcd(n,t1)]; od:
ans;

smd[n_]:=Module[{m=1},While[Mod[m(m+1),n]!=0,m++];GCD[n,m]]; Array[smd,110] (* Harvey P. Dale, Jan 07 2022 *)

f(n) = my(m=1); while ((m*(m+1)) % n, m++); m; \\ A344005
a(n) = gcd(n,f(n)); \\ Michel Marcus, Aug 06 2021
(Python 3.8+)
from math import gcd, prod
from itertools import combinations
from sympy import factorint
from sympy.ntheory.modular import crt
def A345992(n):
    if n == 1:
        return 1
    plist = tuple(p**q for p, q in factorint(n).items())
    return 1 if len(plist) == 1 else gcd(n,int(min(min(crt((m, n//m), (0, -1))[0], crt((n//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, len(plist)//2+1) for d in combinations(plist, l))))) # Chai Wah Wu, Jun 01 2022

a(n) = gcd(n, A182665(n)) = gcd(A182665(n), A344005(n)). - Antti Karttunen, Jun 13 2022

cp's OEIS Frontend

A005384 Sophie Germain primes p: 2p+1 is also prime.

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A002476 Primes of the form 6m + 1.

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A003627 Primes of the form 3n-1.

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A075677 Reduced Collatz function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (3k+1)/2^r, with r as large as possible.

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A039949 Primes of the form 30n - 13.

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