A045326 -id:A045326 - cp's OEIS frontend

A002145 Primes of the form 4*k + 3.

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 563, 571

Offset: 1

N. J. A. Sloane

Or, odd primes p such that -1 is not a square mod p, i.e., the Legendre symbol (-1/p) = -1. [LeVeque I, p. 66]. - N. J. A. Sloane, Jun 28 2008
Primes which are not the sum of two squares, see the comment in A022544. - Artur Jasinski, Nov 15 2006
Natural primes which are also Gaussian primes. (It is a common error to refer to this sequence as "the Gaussian primes".)
Inert rational primes in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
Numbers n such that the product of coefficients of (2n)-th cyclotomic polynomial equals -1. - Benoit Cloitre, Oct 22 2002
For p and q both belonging to the sequence, exactly one of the congruences x^2 = p (mod q), x^2 = q (mod p) is solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Also primes p that divide L((p-1)/2) or L((p+1)/2), where L(n) = A000032(n), the Lucas numbers. Union of A122869 and A122870. - Alexander Adamchuk, Sep 16 2006
Also odd primes p that divide ((p-1)!! + 1) or ((p-2)!! + 1). - Alexander Adamchuk, Nov 30 2006
Also odd primes p that divide ((p-1)!! - 1) or ((p-2)!! - 1). - Alexander Adamchuk, Apr 18 2007
This sequence is a proper subset of the set of the absolute values of negative fundamental discriminants (A003657). - Paul Muljadi, Mar 29 2008
Bernard Frénicle de Bessy discovered that such primes cannot be the hypotenuse of a Pythagorean triangle in opposition to primes of the form 4*n+1 (see A002144). - after Paul Curtz, Sep 10 2008
A079261(a(n)) = 1; complement of A145395. - Reinhard Zumkeller, Oct 12 2008
Subsequence of A007970. - Reinhard Zumkeller, Jun 18 2011
A151763(a(n)) = -1.
Primes p such that p XOR 2 = p - 2. Brad Clardy, Oct 25 2011 (Misleading in the sense that this is a formula for the super-sequence A004767. - R. J. Mathar, Jul 28 2014)
It appears that each term of A004767 is the mean of two terms of this subsequence of primes therein; cf. A245203. - M. F. Hasler, Jul 13 2014
Numbers n > 2 such that ((n-2)!!)^2 == 1 (mod n). - Thomas Ordowski, Jul 24 2016
Odd numbers n > 1 such that ((n-1)!!)^2 == 1 (mod n). - Thomas Ordowski, Jul 25 2016
Primes p such that (p-2)!! == (p-3)!! (mod p). - Thomas Ordowski, Jul 28 2016
See Granville and Martin for a discussion of the relative numbers of primes of the form 4k+1 and 4k+3. - Editors, May 01 2017
Sometimes referred to as Blum primes for their connection to A016105 and the Blum Blum Shub generator. - Charles R Greathouse IV, Jun 14 2018
Conjecture: a(n) for n > 4 can be written as a sum of 3 primes of the form 4k+1, which would imply that primes of the form 4k+3 >= 23 can be decomposed into a sum of 6 nonzero squares. - Thomas Scheuerle, Feb 09 2023

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.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 146-147.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 252.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 66.
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).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 90.

Zak Seidov, 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].
Dario Alpern, Gaussian primes.
Lenore Blum, Manuel Blum, and Mike Shub, A simple unpredictable pseudo-random number generator, SIAM Journal on Computing 15:2 (1 May 1986), pp. 364-383.
Andrew Granville and Greg 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.
Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, On a Dynamical Approach to Some Prime Number Sequences, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018.
E. T. Ordman, Tables of the class number for negative prime discriminants, Deposited in Unpublished Mathematical Table file of Math. Comp. [Annotated scanned partial copy with notes]
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 67.
Harry J. Smith, Gaussian Primes.
Ian Stewart, The Great Mathematical Problems, 2013.
Eric Weisstein's World of Mathematics, Gaussian Prime.
Eric Weisstein's World of Mathematics, Gaussian Integer.
Wolfram Research, The Gauss Reciprocity Law.
Index entries for Gaussian integers and primes.
Index to sequences related to decomposition of primes in quadratic fields.

Cf. A000032, A002144, A003657, A085992, A122869, A122870, A334912.
Cf. A000408, A005098, A095278, A016754.
Apart from initial term, same as A045326.
Cf. A016105.
Cf. A004614 (multiplicative closure).

a002145 n = a002145_list !! (n-1)
a002145_list = filter ((== 1) . a010051) [3, 7 ..]
-- Reinhard Zumkeller, Aug 02 2015, Sep 23 2011

[4*n+3 : n in [0..142] | IsPrime(4*n+3)]; // Arkadiusz Wesolowski, Nov 15 2013

A002145 := proc(n)
    option remember;
    if n = 1 then
        3;
    else
        a := nextprime(procname(n-1)) ;
        while a mod 4 <>  3 do
            a := nextprime(a) ;
        end do;
        return a;
    end if;
end proc:
seq(A002145(n),n=1..20) ; # R. J. Mathar, Dec 08 2011

Select[4Range[150] - 1, PrimeQ] (* Alonso del Arte, Dec 19 2013 *)
Select[ Prime@ Range[2, 110], Length@ PowersRepresentations[#^2, 2, 2] == 1 &] (* or *)
Select[ Prime@ Range[2, 110], JacobiSymbol[-1, #] == -1 &] (* Robert G. Wilson v, May 11 2014 *)

forprime(p=2,1e3,if(p%4==3,print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

def A002145_list(n): return [p for p in prime_range(1, n + 1) if p % 4 == 3]  # Peter Luschny, Jul 29 2014

Remove from A000040 terms that are in A002313.
Intersection of A000040 and A004767. - Alonso del Arte, Apr 22 2014
From Vaclav Kotesovec, Apr 30 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = A243379.
Product_{k>=1} (1 + 1/a(k)^2) = A243381.
Product_{k>=1} (1 - 1/a(k)^3) = A334427.
Product_{k>=1} (1 + 1/a(k)^3) = A334426.
Product_{k>=1} (1 - 1/a(k)^4) = A334448.
Product_{k>=1} (1 + 1/a(k)^4) = A334447.
Product_{k>=1} (1 - 1/a(k)^5) = A334452.
Product_{k>=1} (1 + 1/a(k)^5) = A334451. (End)
From Vaclav Kotesovec, May 05 2020: (Start)
Product_{k>=1} (1 + 1/a(k)) / (1 + 1/A002144(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/a(k)) / (1 - 1/A002144(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log(2 * (2^(n*s) - 1) * (n*s - 1)! * zeta(n*s) / (Pi^(n*s) * abs(EulerE(n*s - 1))))/n, s >= 3 odd number. - Dimitris Valianatos, May 20 2020

More terms from James Sellers, Apr 21 2000

A001348 Mersenne numbers: 2^p - 1, where p is prime.

Original entry on oeis.org

3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607, 536870911, 2147483647, 137438953471, 2199023255551, 8796093022207, 140737488355327, 9007199254740991, 576460752303423487, 2305843009213693951, 147573952589676412927, 2361183241434822606847

Offset: 1

N. J. A. Sloane

Mersenne numbers A000225 whose indices are primes. - Omar E. Pol, Aug 31 2008
All terms are of the form 4k-1. - Paul Muljadi, Jan 31 2011
Smallest number with Hamming weight A000120 = prime(n). - M. F. Hasler, Oct 16 2018
The 5th, 9th, 10th, ... terms are not prime. See A000668 and A065341 for the primes and for the composites in this sequence. - M. F. Hasler, Nov 14 2018 [corrected by Jerzy R Borysowicz, Apr 08 2025]
Except for the first term 3: all prime factors of 2^p-1 must be 1 or -1 (mod 8), and 1 (mod 2p). - William Hu, Mar 10 2024

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 16.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
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).

T. D. Noe, Table of n, a(n) for n = 1..100
Raymond Clare Archibald, Mersenne's Numbers, Scripta Mathematica, Vol. 3 (1935), pp. 112-119.
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and S. S. Wagstaff, Jr., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
C. K. Caldwell, Mersenne Primes
Will Edgington, Mersenne Page> [from Internet Archive Wayback Machine].
Graham Everest, Shaun Stevens, Duncan Tamsett and Tom Ward, Primes generated by recurrence sequences, Amer. Math. Monthly, Vol. 114, No. 5 (2007), pp. 417-431.
Paul Garrett, Lucas-Lehmer criterion for primality of Mersenne numbers, 2010.
Jiří Klaška, A Simple Proof of Skula's Theorem on Prime Power Divisors of Mersenne Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.4.3.
Gabriel Lapointe, On finding the smallest happy numbers of any heights, arXiv:1904.12032 [math.NT], 2019.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
Anthony G. Shannon, Hakan Akkuş, Yeşim Aküzüm, Ömür Deveci, and Engin Özkan, A partial recurrence Fibonacci link, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 530-537. See Table 1, p. 531.
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Thesaurus.maths.org, Mersenne Number.
Gérard Villemin's Almanach of Numbers, Nombre de Mersenne.
Eric Wegrzynowski, Nombres de Mersenne. [from Internet Archive Wayback Machine]
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., Vol. 3 (1892), pp. 265-284.

Cf. A000043, A000668, A046051, A057951-A057958, A100105, A184085, A262153.
Cf. A000040, A000225. - Omar E. Pol, Aug 31 2008
Cf. A002145, A045326.

[2^NthPrime(n)-1: n in [1..30]]; // Vincenzo Librandi, Feb 04 2016

A001348 := n -> 2^(ithprime(n))-1: seq (A001348(n), n=1..18);

Table[2^Prime[n]-1, {n, 20}] (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

a(n)=1<Charles R Greathouse IV, Jun 10 2011

from sympy import prime
def a(n): return 2**prime(n)-1
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Mar 28 2022

a(n) = 2^A000040(n) - 1, n >= 1. - Wolfdieter Lang, Oct 26 2014
a(n) = A000225(A000040(n)). - Omar E. Pol, Aug 31 2008
A000668(n) = a(A016027(n)). - Omar E. Pol, Jun 29 2012
Sum_{n>=1} 1/a(n) = A262153. - Amiram Eldar, Nov 20 2020
Product_{n>=1} (1 - 1/a(n)) = A184085. - Amiram Eldar, Nov 22 2022

A004144 Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 103, 107, 108, 112, 114, 118, 121, 124, 126, 127

Offset: 1

N. J. A. Sloane

nonn

Also numbers with no prime factors of form 4*k+1.
m is a term iff A072438(m) = m.
Density 0. - Charles R Greathouse IV, Apr 16 2012
Closed under multiplication. Primitive elements are A045326, 2 and the primes of form 4*k+3. - Jean-Christophe Hervé, Nov 17 2013

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Evan M. Bailey, Table of n, a(n) for n = 1..20000 (Terms 1..1000 from T. D. Noe)
Evan M. Bailey, a004144.cpp.
Steven R. Finch, Landau-Ramanujan Constant [Broken link]
Steven R. Finch, Landau-Ramanujan Constant [From the Wayback machine]
Daniel Shanks, Non-hypotenuse numbers, Fib. Quart., Vol. 13, No. 4 (1975), pp. 319-321.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for sequences related to sums of squares.

Complement of A009003.
Cf. A000290, A002145, A005089, A072437, A244659, A244662.
The subsequence of primes is A045326.

import Data.List (elemIndices)
a004144 n = a004144_list !! (n-1)
a004144_list = map (+ 1) $ elemIndices 0 a005089_list
-- Reinhard Zumkeller, Jan 07 2013

fQ[n_] := If[n > 1, First@ Union@ Mod[ First@# & /@ FactorInteger@ n, 4] != 1, True]; Select[ Range@ 127, fQ]
A004144 = Select[Range[127],Length@Reduce[s^2 + t^2 == s # && s > t > 0, Integers] == 0 &] (* Gerry Martens, Jun 09 2020 *)

is(n)=n==1||vecmin(factor(n)[,1]%4)>1 \\ Charles R Greathouse IV, Apr 16 2012

list(lim)=my(v=List(),u=vectorsmall(lim\=1)); forprimestep(p=5,lim,4, forstep(n=p,lim,p, u[n]=1)); for(i=1,lim, if(u[i]==0, listput(v,i))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022

A005089(a(n)) = 0. - Reinhard Zumkeller, Jan 07 2013

The number of terms below x is ~ (A * x / sqrt(log(x))) * (1 + C/log(x) + O(1/log(x)^2)), where A = A244659 and C = A244662 (Shanks, 1975). - Amiram Eldar, Jan 29 2022

More terms from Reinhard Zumkeller, Jun 17 2002

Name clarified by Evan M. Bailey, Sep 17 2019

A062327 Number of divisors of n over the Gaussian integers.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 2, 7, 3, 12, 2, 10, 4, 6, 8, 9, 4, 9, 2, 20, 4, 6, 2, 14, 9, 12, 4, 10, 4, 24, 2, 11, 4, 12, 8, 15, 4, 6, 8, 28, 4, 12, 2, 10, 12, 6, 2, 18, 3, 27, 8, 20, 4, 12, 8, 14, 4, 12, 2, 40, 4, 6, 6, 13, 16, 12, 2, 20, 4, 24, 2, 21, 4, 12, 18, 10, 4, 24, 2, 36, 5, 12, 2, 20, 16, 6

Offset: 1

Reiner Martin, Jul 12 2001

Divisors which are associates are identified (two Gaussian integers z1, z2 are associates if z1 = u * z2 where u is a unit, i.e., one of 1, i, -1, -i).
a(A004614(n)) = A000005(n). - Vladeta Jovovic, Jan 23 2003
a(A004613(n)) = A000005(n)^2. - Benedikt Otten, May 22 2013

			For example, 5 has divisors 1, 1+2i, 2+i and 5.

Cf. A027748, A124010.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): this sequence ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319442.

a062327 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e                  = 2 * e + 1
   f p e | p `mod` 4 == 1 = (e + 1) ^ 2
         | otherwise      = e + 1
-- Reinhard Zumkeller, Oct 18 2011

a:= n-> mul(`if`(i[1]=2, 2*i[2]+1, `if`(irem(i[1], 4)=3,
                 i[2]+1, (i[2]+1)^2)), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 09 2021

Table[Length[Divisors[n, GaussianIntegers -> True]], {n, 30}] (* Alonso del Arte, Jan 25 2011 *)
DivisorSigma[0,Range[90],GaussianIntegers->True] (* Harvey P. Dale, Mar 19 2017 *)

a(n)=
{
    my(r=1,f=factor(n));
    for(j=1,#f[,1], my(p=f[j,1],e=f[j,2]);
        if(p==2,r*=(2*e+1));
        if(p%4==1,r*=(e+1)^2);
        if(p%4==3,r*=(e+1));
    );
    return(r);
}  \\ Joerg Arndt, Dec 09 2016

Presumably a(n) = 2 iff n is a rational prime == 3 mod 4 (see A045326). - N. J. A. Sloane, Jan 07 2003, Feb 23 2007

Multiplicative with a(2^e) = 2*e+1, a(p^e) = e+1 if p mod 4=3 and a(p^e) = (e+1)^2 if p mod 4=1. - Vladeta Jovovic, Jan 23 2003

A100672 a(1) = 1; thereafter, a(n) = 1 if n-th prime is 3 mod 4, 0 if n-th prime is 1 mod 4.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 06 2004

Second least-significant bit in the binary expansion of the n-th prime.

a(n)=1 iff prime(n) is a member of A045326 (equivalently for n>1, iff prime(n)-3 is divisible by 4).

			a(2)=1 because prime(2)=11_2 (in binary; decimal = 3_10) and its 2^1 bit is 1.
a(3)=0 because prime(3)=101_2 (in binary; decimal = 5_10) and its 2^1 bit is 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.
Eric Weisstein's World of Mathematics, Gaussian Prime.

Cf. A000040, A045326, A002144, A002145, A039702.

RUNS transform is essentially A091237.

A100672 := proc(n)
        if n = 1 then
                1 ;
        else
                ((ithprime(n) mod 4)-1)/2;
        end if;
end proc: # R. J. Mathar, Oct 06 2011

Table[Reverse[RealDigits[Prime[k], 2][[1]]][[2]], {k, 1, 128}]

for(k=1,105,print1( bittest(prime(k), 1), ", ")) \\ Washington Bomfim, Jan 18 2011

from sympy import prime
def A100672(n): return int(prime(n)>>1&1) # Chai Wah Wu, Jun 23 2023

a(n) = 1-A098033(n), n>1. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008

a(n) = floor(prime(n)/2) mod 2. - Alois P. Heinz, Jul 16 2024

Edxited by N. J. A. Sloane, Jan 11 2025

A167134 Primes congruent to {2, 3, 5, 7} mod 11.

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 47, 71, 73, 79, 101, 113, 137, 139, 157, 167, 179, 181, 211, 223, 227, 233, 269, 271, 277, 293, 311, 313, 337, 359, 379, 401, 409, 421, 431, 443, 467, 487, 491, 509, 541, 557, 563, 577, 599, 601, 607, 619, 641, 643, 673, 709, 733, 739, 751

Offset: 1

Klaus Brockhaus, Oct 28 2009

Primes p such that p mod 11 is prime.
Primes of the form 11*n+r where n >= 0 and r is in {2, 3, 5, 7}.
2 and primes congruent to {3, 5, 7, 13} mod 22. - Chai Wah Wu, Apr 29 2025

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

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.

[ p: p in PrimesUpTo(760) | p mod 11 in {2, 3, 5, 7} ];
[ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 11] | exists(u){ r: r in {2, 3, 5,7} | p eq (11*n+r) } } ];

Select[Prime[Range[600]],MemberQ[{2, 3, 5, 7},Mod[#,11]]&] (* Vincenzo Librandi, Aug 05 2012 *)

A167135 Primes congruent to {2, 3, 5, 7, 11} mod 12.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 331, 347, 353, 359

Offset: 1

Klaus Brockhaus, Oct 28 2009

Primes p such that p mod 12 is prime.
Primes of the form 12*n+r where n >= 0 and r is in {2, 3, 5, 7, 11}.
Except for the prime 2, these are the primes that are encountered in the set of numbers {x, f(f(x))} where x is of the form 4k+3 with k>=0, and where f(x) is the 3x+1-problem function, and f(f(x)) the second iteration value. Indeed this sequence is the set union of 2 and A002145 (4k+3 primes) and A007528 (6k+5 primes), since f(f(4k+3))=6k+5. Equivalently one does not get any prime from A068228 (the complement of the present sequence). - Michel Marcus and Bill McEachen, May 07 2016

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

Subsequences: A002145, A007528. Complement: A068228.

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.

[ p: p in PrimesUpTo(760) | p mod 12 in {2, 3, 5, 7, 11} ];

[ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 12] | exists(u){ r: r in {2, 3, 5,7, 11} | p eq (12*n+r) } } ];

isA167135  := n -> isprime(n) and not modp(n, 12) != 1:
select(isA167135, [$1..360]); # Peter Luschny, Mar 28 2018

Select[Prime[Range[400]],MemberQ[{2,3, 5, 7, 11},Mod[#,12]]&] (* Vincenzo Librandi, Aug 05 2012 *)
Select[Prime[Range[72]], Mod[#, 12] != 1 &] (* Peter Luschny, Mar 28 2018 *)

A270109 a(n) = n^3 + (n+1)*(n+2).

Original entry on oeis.org

2, 7, 20, 47, 94, 167, 272, 415, 602, 839, 1132, 1487, 1910, 2407, 2984, 3647, 4402, 5255, 6212, 7279, 8462, 9767, 11200, 12767, 14474, 16327, 18332, 20495, 22822, 25319, 27992, 30847, 33890, 37127, 40564, 44207, 48062, 52135, 56432, 60959, 65722, 70727, 75980, 81487, 87254

Offset: 0

Bruno Berselli, Mar 11 2016, at the suggestion of Giuseppe Amoruso in BASE Cinque forum

For n>1, many consecutive terms of the sequence are generated by floor(sqrt(n^2 + 2)^3) + n^2 + 2.
It appears that this is a subsequence of A000037 (the nonsquares).
The primes in the sequence belong to A045326.
Inverse binomial transform is 2, 5, 8, 6, 0, 0, 0, ... (0 continued).

Bruno Berselli, Table of n, a(n) for n = 0..1000
MathsSmart, Number pattern and Puzzle - 7, 20, 47, 94, 167.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Subsequence of A001651, A047212.
Cf. A000037, A045326.
Cf. A027444: numbers of the form n^3+n*(n+1); A085490: numbers of the form n^3+(n-1)*n.
Cf. A008865: numbers of the form n+(n+1)*(n+2); A130883: numbers of the form n^2+(n+1)*(n+2).

Magma
```
[n^3+(n+1)*(n+2): n in [0..50]];
```

Table[n^3 + (n + 1) (n + 2), {n, 0, 50}]

Maxima
```
makelist(n^3+(n+1)*(n+2), n, 0, 50);
```
PARI
```
vector(50, n, n--; n^3+(n+1)*(n+2))
```
Sage
```
[n^3+(n+1)*(n+2) for n in (0..50)]
```

O.g.f.: (2 - x + 4*x^2 + x^3)/(1 - x)^4.
E.g.f.: (2 + x)*(1 + x)^2*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3.
a(n+h) - a(n) + a(n-h) = n^3 + n^2 + (6*h^2+3)*n + (2*h^2+2) for any h. This identity becomes a(n) = n^3 + n^2 + 3*n + 2 if h=0.
a(h*a(n) + n) = (h*a(n))^3 + (3*n+1)*(h*a(n))^2 + (3*n^2+2*n+3)*(h*a(n)) + a(n) for any h, therefore a(h*a(n) + n) is always a multiple of a(n).
a(n) + a(-n) = 2*A059100(n) = A255843(n).
a(n) - a(-n) = 4*A229183(n).

A167119 Primes congruent to 2, 3, 5, 7 or 11 (mod 13).

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 31, 37, 41, 59, 67, 83, 89, 107, 109, 137, 163, 167, 193, 197, 211, 223, 239, 241, 263, 271, 293, 317, 349, 353, 367, 379, 397, 401, 419, 421, 431, 449, 457, 479, 499, 509, 523, 557, 577, 587, 601, 613, 631, 653, 661, 683, 691, 709, 733, 739, 743, 757

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 27 2009

Primes which have a remainder mod 13 that is prime.

Union of A141858, A100202, A102732, A140371 and A140373. - R. J. Mathar, Oct 29 2009

			11 mod 13 = 11, 29 mod 13 = 3, 31 mod 13 = 5, hence 11, 29 and 31 are in the sequence.

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

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135: primes p such that p mod k is prime, for k = 3..12 resp.

[ p: p in PrimesUpTo(740) | p mod 13 in {2, 3, 5, 7, 11} ]; // Klaus Brockhaus, Oct 28 2009

f[n_]:=PrimeQ[Mod[n,13]]; lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6,6!}];lst
Select[Prime[Range[4000]],MemberQ[{2, 3, 5, 7, 11},Mod[#,13]]&] (* Vincenzo Librandi, Aug 05 2012 *)

{forprime(p=2, 740, if(isprime(p%13), print1(p, ",")))} \\ Klaus Brockhaus, Oct 28 2009

Edited by Klaus Brockhaus and R. J. Mathar, Oct 28 2009 and Oct 29 2009

cp's OEIS Frontend

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