A045331 -id:A045331 - cp's OEIS frontend

A002476 Primes of the form 6m + 1.

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

A002313 Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617

Offset: 1

N. J. A. Sloane

Or, primes p such that x^2 - p*y^2 represents -1.
Primes which are not Gaussian primes (meaning not congruent to 3 mod 4).
Every Fibonacci prime (with the exception of F(4) = 3) is in the sequence. If p = 2n+1 is the prime index of the Fibonacci prime, then F(2n+1) = F(n)^2 + F(n+1)^2 is the unique representation of the prime as sum of two squares. - Sven Simon, Nov 30 2003
Except for 2, primes of the form x^2 + 4y^2. See A140633. - T. D. Noe, May 19 2008
Primes p such that for all p > 2, p XOR 2 = p + 2. - Brad Clardy, Oct 25 2011
Greatest prime divisor of r^2 + 1 for some r. - Michel Lagneau, Sep 30 2012
Empirical result: a(n), as a set, compose the prime factors of the family of sequences produced by A005408(j)^2 + A005408(j+k)^2 = (2j+1)^2 + (2j+2k+1)^2, for j >= 0, and a given k >= 1 for each sequence, with the addition of the prime factors of k if not already in a(n). - Richard R. Forberg, Feb 09 2015
Primes such that when r is a primitive root then p-r is also a primitive root. - Emmanuel Vantieghem, Aug 13 2015
Primes of the form (x^2 + y^2)/2. Note that (x^2 + y^2)/2 = ((x+y)/2)^2 + ((x-y)/2)^2 = a^2 + b^2 with x = a + b and y = a - b. More generally, primes of the form (x^2 + y^2) / A001481(n) for every fixed n > 1. - Thomas Ordowski, Jul 03 2016
Numbers n such that ((n-2)!!)^2 == -1 (mod n). - Thomas Ordowski, Jul 25 2016
Primes p such that (p-1)!! == (p-2)!! (mod p). - Thomas Ordowski, Jul 28 2016
The product of 2 different terms (x^2 + y^2)(z^2 + v^2) = (xz + yv)^2 + (xv - yz)^2 is sum of 2 squares (A000404) because (xv - yz)^2 > 0. If x were equal to yz/v then (x^2 + y^2)/(z^2 + v^2) would be equal to ((yz/v)^2 + y^2)/(z^2 + v^2) = y^2/v^2 which is not possible because (x^2 + y^2) and (z^2 + v^2) are prime numbers. For example, (2^2 + 5^2)(4^2 + 9^2) = (2*4 + 5*9)^2 + (2*9 - 5*4)^2. - Jerzy R Borysowicz, Mar 21 2017

			13 is in the sequence since it is prime and 13 = 4*3 + 1.  Also 13 = 2^2 + 3^2.  And -1 is a square (mod 13): -1 + 2*13 = 25 = 5^2.  Of course, only the first term is congruent to 2 (mod 4). - _Michael B. Porter_, Jul 04 2016

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 872.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 251, 252.
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).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Dario Alpern, Online program that calculates sum of squares representation
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
Eric Weisstein's World of Mathematics, Fermat's 4n+1 Theorem
G. Xiao, Two squares
Index entries for Gaussian integers and primes

Apart from initial term, same as A002144. For values of x and y see A002330 and A002331.

Cf. A033203, A038873, A038874, A045331, A008784, A057129, A084163, A084165, A137351.

a002313 n = a002313_list !! (n-1)
a002313_list = filter ((`elem` [1,2]) . (`mod` 4)) a000040_list
-- Reinhard Zumkeller, Feb 04 2014

[p: p in PrimesUpTo(700) | p mod 4 in {1,2}]; // Vincenzo Librandi, Feb 18 2015

with(numtheory): for n from 1 to 300 do if ithprime(n) mod 4 = 1 or ithprime(n) mod 4 = 2 then printf(`%d,`,ithprime(n)) fi; od:
# alternative
A002313 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        2;
    elif n = 2 then
        5;
    else
        for a from procname(n-1)+4 by 4 do
            if isprime(a) then
                return a ;
            end if;
        end do:
    end if;
end proc:
seq(A002313(n),n=1..100) ; # R. J. Mathar, Feb 01 2024

Select[ Prime@ Range@ 115, Mod[#, 4] != 3 &] (* Robert G. Wilson v *)
fQ[n_] := Solve[x^2 + 1 == n*y^2, {x, y}, Integers] == {}; Select[ Prime@ Range@ 115, fQ] (* Robert G. Wilson v, Dec 19 2013 *)

select(p->p%4!=3, primes(1000)) \\ Charles R Greathouse IV, Feb 11 2011

a(n) ~ 2n log n. - Charles R Greathouse IV, Jul 04 2016

a(n) = A002331(n)^2 + A002330(n)^2. See crossrefs. - Wolfdieter Lang, Dec 11 2016

More terms from Henry Bottomley, Aug 10 2000

More terms from James Sellers, Aug 22 2000

A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

Original entry on oeis.org

3, 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

Offset: 1

N. J. A. Sloane, Mira Bernstein and Robert G. Wilson v

Also, odd primes p such that -3 is a square mod p. - N. J. A. Sloane, Dec 25 2017
Equivalently, primes of the form p = (x^3 - y^3)/(x - y). If x=y+1 we get the cuban primes A002407, which is therefore a subsequence.
These are not to be confused with the Eisenstein primes, which are the primes in the ring of integers Z[w], where w = (-1+sqrt(-3))/2. The present sequence gives the rational primes which are also Eisenstein primes. - N. J. A. Sloane, Feb 06 2008
Also primes of the form x^2+3y^2 and, except for 3, x^2+xy+7y^2. See A140633. - T. D. Noe, May 19 2008
Conjecture: this sequence is Union(A002383,A162471). - Daniel Tisdale, Jul 04 2009
Primes p such that antiharmonic mean B(p) of the numbers k < p such that gcd(k, p) = 1 is not integer, where B(p) = A053818(p) / A023896(p) = A175505(p) / A175506(p) = (2p - 1) / 3. Primes p such that A175506(p) > 1. Subsequence of A179872. Union a(n) + A179891 = A179872. Example: a(6) = 37 because B(37) = A053818(37) / A023896(37) = A175505(37) / A175506(37) = 16206 / 666 = 73 / 3 (not integer). Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A003627, A034934. - Jaroslav Krizek, Aug 01 2010
Subsequence of Loeschian numbers, cf. A003136 and A024614; A088534(a(n)) > 0. - Reinhard Zumkeller, Oct 30 2011
Primes such that there exist a unique x, y, with 1 < x <= y < p, x + y == 1 (mod p) and x * y == 1 (mod p). - Jon Perry, Feb 02 2014
The prime factors of A002061. - Richard R. Forberg, Dec 10 2014
This sequence gives the primes p which solve s^2 == -3 (mod 4*p) (see Buell, Proposition 4.1., p. 50, for Delta = -3). p = 2 is not a solution. x^2 == -3 (mod 4) has solutions for all odd x. x^2 == -3 (mod  p) has for odd primes p, not 3, the solutions of Legendre(-3|p) = +1 which are p == {1, 7} (mod 12). For p = 3 the representative solution is x = 0. Hence the solution of s^2 == -3 (mod 4*p) are the odd primes p = 3 and p == {1, 7} (mod 12) (or the primes p = 0, 1 (mod 3)). - Wolfdieter Lang, May 22 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 50.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

T. D. Noe, Table of n, a(n) for n = 1..1000
U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Eisenstein Integer.

Subsequence of A003136.
Subsequences include A002407, A002648, and A201477.
Apart from initial term, same as A045331.
Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2).
Cf. A000040, A003627.
Primes in A003136 and A034017.

a007645 n = a007645_list !! (n-1)
a007645_list = filter ((== 1) . a010051) $ tail a003136_list
-- Reinhard Zumkeller, Jul 11 2013, Oct 30 2011

select(isprime,[3, seq(6*k+1, k=1..1000)]); # Robert Israel, Dec 12 2014

Join[{3},Select[Prime[Range[150]],Mod[#,3]==1&]] (* Harvey P. Dale, Aug 21 2021 *)

forprime(p=2,1e3,if(p%3<2,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

p == 0 or 1 (mod 3).
{3} UNION A002476. - R. J. Mathar, Oct 28 2008
A007645 UNION A003627 = A000040. - Juri-Stepan Gerasimov, Jan 28 2010

Entry revised by N. J. A. Sloane, Jan 29 2013

A038874 Primes p such that 3 is a square mod p.

Original entry on oeis.org

2, 3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601

Offset: 1

N. J. A. Sloane

Also primes congruent to {1, 2, 3, 11} mod 12.
The subsequence p = 1 (mod 4) corresponds to A068228 and only these entries of a(n) are squares mod 3 (from the quadratic reciprocity law). - Lekraj Beedassy, Jul 21 2004
Largest prime factors of n^2 - 3. - Vladimir Joseph Stephan Orlovsky, Aug 12 2009
Aside from 2 and 3, primes p such that Legendre(3, p) = 1. Bolker asserts there are infinitely many of these primes. - Alonso del Arte, Nov 25 2015
The associated bases of the squares are 1, 0, 5, 4, 7, 15, 12, 11, 8, 28, 21, 13...: 1^2 = 3 -1*2, 0^2 = 3-1*3, 5^2 = 3+ 2*11, 4^2 = 3+1*13, 7^2 = 3+2*23, 15^2 = 3+6*37, 12^2 = 3+3*47,... - R. J. Mathar, Feb 23 2017

			11 is in the sequence since the equation x^2 - 11y = 3 has solutions, such as x = 5, y = 2.
13 is in the sequence since the equation x^2 - 13y = 3 has solutions, such as x = 4, y = 1.
17 is not in the sequence because x^2 - 17y = 3 has no solutions in integers; Legendre(3, 17) = -1.

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 74, Theorem 25.3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Tamara M. Lavshuk, Regular polygons and polyhedra over finite field, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.

Cf. A002313, A033203, A038873, A045331, A057125.

If the first two terms are omitted we get A097933. A040101 is another sequence.

[p: p in PrimesUpTo(1200) | p mod 12 in [1, 2, 3, 11]]; // Vincenzo Librandi, Aug 08 2012

select(isprime, [2,3, seq(seq(6+s+12*i, s=[-5,5]),i=0..1000)]); # Robert Israel, Dec 23 2015

Select[Prime[Range[250]], MemberQ[{1, 2, 3, 11}, Mod[#, 12]] &] (* Vincenzo Librandi, Aug 08 2012 *)
Select[Flatten[Join[{2, 3}, Table[{12n - 1, 12n + 1}, {n, 50}]]], PrimeQ] (* Alonso del Arte, Nov 25 2015 *)

forprime(p=2, 1e3, if(issquare(Mod(3, p)), print1(p , ", "))) \\ Altug Alkan, Dec 04 2015

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

More terms from Henry Bottomley, Aug 10 2000

A045375 Primes congruent to {1, 2} mod 6.

Original entry on oeis.org

2, 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, 631, 643, 661, 673, 691

Offset: 1

N. J. A. Sloane

nonn

Apart from initial term, same as A002476 = A007645 \ {2} = A045331 \ {2,3}. - M. F. Hasler, Apr 25 2008

Primes of the form 6*m - 3/2 -+ 5/2. A045375 UNION A045410 = A000040. - Juri-Stepan Gerasimov, Jan 28 2010

Cf. A002476, A007645, A045331.

Cf. A000040 (the primes), A045410 (the primes of the form 6*k-2-+1). - Juri-Stepan Gerasimov, Jan 28 2010

[p: p in PrimesUpTo(740)|p mod 6 in {1,2}]; // Vincenzo Librandi, Dec 18 2010

Select[Prime[Range[150]], MemberQ[{1, 2}, Mod[#, 6]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2012 *)

More terms from Vincenzo Librandi, Dec 18 2010

A057128 Numbers n such that -3 is a square mod n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 12, 13, 14, 19, 21, 26, 28, 31, 37, 38, 39, 42, 43, 49, 52, 57, 61, 62, 67, 73, 74, 76, 78, 79, 84, 86, 91, 93, 97, 98, 103, 109, 111, 114, 122, 124, 127, 129, 133, 134, 139, 146, 147, 148, 151, 156, 157, 158, 163, 169, 172, 181, 182, 183, 186, 193

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

The fact that there are no numbers in this sequence of the form 6k+5 leads to the result that all prime factors of central polygonal numbers (A002061 of the form n^2-n+1) are either 3 or of the form 6k+1. This in turn leads to there being an infinite number of primes of the form 6k+1, since if P=product[all known primes of form 6k+1] then all the prime factors of 9P^2-3P+1 must be unknown primes of form 6k+1.
Numbers that are not multiples of 8 or 9 and for which all prime factors greater than 3 are congruent to 1 mod 6. - Eric M. Schmidt, Apr 21 2013
Numbers that divide at least some member of A117950. - Robert Israel, Feb 19 2016

			a(7)=13 since -3 mod 13=10 mod 13=6^2 mod 13.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Includes the primes in A045331 and these (primes congruent to {1, 2, 3} mod 6) are the prime factors of the terms in this sequence. Cf. A008784, A057125, A057126, A057127, A057129.

Cf. A117950.

select(t -> numtheory:-quadres(-3,t) = 1, {$1..1000}); # Robert Israel, Feb 19 2016

Select[Range[200], IntegerQ[PowerMod[-3, 1/2, #]]&] // Quiet (* Jean-François Alcover, Mar 05 2019 *)

isok(n) = issquare(Mod(-3,n)); \\ Michel Marcus, Feb 19 2016

def A057128(n) :
    if n%8==0 or n%9==0: return False
    for (p, m) in factor(n) :
        if p % 6 not in [1, 2, 3] : return False
        return True
# Eric M. Schmidt, Apr 21 2013

A047246 Numbers that are congruent to {0, 1, 2, 3} mod 6.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 12, 13, 14, 15, 18, 19, 20, 21, 24, 25, 26, 27, 30, 31, 32, 33, 36, 37, 38, 39, 42, 43, 44, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 61, 62, 63, 66, 67, 68, 69, 72, 73, 74, 75, 78, 79, 80, 81, 84, 85, 86, 87, 90, 91, 92, 93, 96, 97, 98

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047238 with A047241. - Guenther Schrack, Feb 12 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A045331 (primes congruent to {1,2,3} mod 6), A047238, A047241.

Complement: A047257.

Filtered([0..100],n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 2 or n mod 6 = 3); # Muniru A Asiru, Feb 20 2019

a047246 n = a047246_list !! (n-1)
a047246_list = [0..3] ++ map (+ 6) a047246_list
-- Reinhard Zumkeller, Jan 15 2013

[Floor((6/5)*Floor(5*(n-1)/4)) : n in [1..100]]; // Wesley Ivan Hurt, May 21 2016

A047246:=n->(6*n-9-I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/4: seq(A047246(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(6n-9-I^(2n)-(1-I)*I^(-n)-(1+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(1+x+x^2+3*x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(1+x+x^2+3*x^3)/((1-x)*(1-x^4))).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x^2*(1+x+x^2+3*x^3) / ((1+x)*(1-x)^2*(1+x^2)). - R. J. Mathar, Oct 08 2011
a(n) = floor((6/5)*floor(5*(n-1)/4)). - Bruno Berselli, May 03 2016
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 9 - i^(2*n) - (1-i)*i^(-n) - (1+i)*i^n)/4 where i=sqrt(-1).
a(2*n) = A047241(n), a(2*n-1) = A047238(n). (End)
E.g.f.: (6 + sin(x) - cos(x) + (3*x - 4)*sinh(x) + (3*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
From Guenther Schrack, Feb 12 2019: (Start)
a(n) = (6*n - 9 - (-1)^n - 2*(-1)^(n*(n+1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=1, a(3)=2, a(4)=3, for n > 4. (End)
Sum_{n>=2} (-1)^n/a(n) = Pi/(6*sqrt(3)) + 2*log(2)/3. - Amiram Eldar, Dec 16 2021
a(n)-a(n-1) = A093148(n-2). - R. J. Mathar, May 01 2024

More terms from Wesley Ivan Hurt, May 21 2016

A235473 Primes whose base-3 representation is also the base-4 representation of a prime.

Original entry on oeis.org

2, 43, 61, 67, 97, 103, 127, 139, 151, 157, 199, 211, 229, 277, 283, 331, 337, 349, 373, 379, 433, 439, 463, 499, 523, 571, 601, 607, 727, 751, 787, 823, 853, 883, 919, 991, 1063, 1087, 1117, 1213, 1249, 1327, 1381, 1429, 1483, 1531, 1567, 1597, 1627, 1759, 1783, 1867, 1999

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

This is a subsequence of A045331 and A045375.

			43 = 1121_3 and 1121_4 = 89 are both prime, so 43 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235266, A235474, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,3],4]]&] (* Harvey P. Dale, Oct 16 2015 *)

is(p,b=4,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

A244146 Primes of the form x^2 + x*y + y^2 with x, y primes.

Original entry on oeis.org

19, 67, 79, 109, 163, 199, 349, 433, 457, 607, 691, 739, 937, 997, 1063, 1093, 1327, 1423, 1447, 1489, 1579, 1753, 1777, 1987, 2017, 2089, 2203, 2287, 2383, 2749, 3229, 3463, 3847, 3943, 4051, 4177, 4513, 4567, 5347, 5413, 5479, 5557, 5653, 6079, 6133, 6271, 6661

Offset: 1

Peter Luschny, Jun 21 2014

nonn

Equally: primes that are of the form (p+q)^2 - p*q, with p, q primes. - Antti Karttunen, Jun 21 2014

			The terms 19, 67, 79 and 1777753 are in the sequence because they can be represented as:
19 = 2^2 + 2*3 + 3^2 = (2+3)^2 - 2*3.
67 = 2^2 + 2*7 + 7^2 = (2+7)^2 - 2*7.
79 = 3^2 + 3*7 + 7^2 = (3+7)^2 - 3*7.
1777753 = 677^2 + 677*859 + 859^2 = (677+859)^2 - 677*859.

Jean-François Alcover, Table of n, a(n) for n = 1..2089

Subsequence of A007645.

Cf. A045331.

Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[Reduce[p == x^2 + x y + y^2, {x, y}, Primes] =!= False, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Jul 12 2019 *)

A240920 Prime numbers that occur as divisors of numbers of the form m^2 + 5.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 127, 149, 163, 167, 181, 223, 227, 229, 241, 263, 269, 281, 283, 307, 347, 349, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487, 503, 509, 521, 523, 541, 547, 563

Offset: 1

J. Lowell, Aug 02 2014

Conjecture: a prime number is in this sequence if and only if its next-to-last digit is even.

The law of quadratic reciprocity shows an odd prime is in the sequence if and only if it is 1, 3, 5, 7 or 9 (mod 20). This proves the above conjecture, so the sequence is the union of {2, 5} and A139513. - Jens Kruse Andersen, Aug 09 2014

			23 is in the sequence because it divides 8^2+5=69 with m=8.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
David Lowry-Duda, Unexpected Conjectures about -5 Modulo Primes, College Mathematics Journal, Vol. 46, No. 1 (Jan 2015), pp.56-57.

Cf. A002313 (k=1 or k=4), A033203 (k=2), A045331 (k=3), A139513.

isA240920 := proc(p)
    local n;
    if isprime(p) then
        for n from 0 to p do
            if modp(n^2+5,p) = 0 then
                return true;
            end if;
        end do:
        false;
    else
        false;
    end if;
end proc:
for i from 1 to 600 do
    p := ithprime(i) ;
    if isA240920(p) then
        printf("%d,",p);
    end if;
end do:

select(p->issquare(Mod(-5,p)), primes(100)) \\ Charles R Greathouse IV, Nov 29 2016

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

cp's OEIS Frontend

A002476 Primes of the form 6m + 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

J

Magma

Maple

Mathematica

PARI

Formula

A002313 Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A038874 Primes p such that 3 is a square mod p.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A045375 Primes congruent to {1, 2} mod 6.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Extensions

A057128 Numbers n such that -3 is a square mod n.

Views

Author