A055664 -id:A055664 - cp's OEIS frontend

A055028 Number of Gaussian primes of norm n.

0, 0, 4, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jun 09 2000

These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).

			There are 8 Gaussian primes of norm 5, +-1 +- 2i and +-2 +- i, but only two inequivalent ones (2 +- i).

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for Gaussian integers and primes

Cf. A055025-A055029, A055664-... , A295996.

A055028 := proc(n::integer)
    local c,a,b ;
    c := 0 ;
    for a from -n to n do
        if issqr(n-a^2) then
            b := sqrt(n-a^2) ;
            if GaussInt[GIprime](a+b*I) and a^2+b^2=n then
                if b = 0 then
                    c := c+1 ; # a+i*b and a-i*b
                else
                    c := c+2 ; # a+i*b and a-i*b
                end if;
            end if;
        end if;
    end do:
    c ;
end proc:
seq( A055028(n),n=0..50) ; # R. J. Mathar, Jul 22 2021

a[n_ /; PrimeQ[n] && Mod[n, 4] == 1] = 8; a[2] = 4; a[n_ /; (p = Sqrt[n]; PrimeQ[p] && Mod[p, 4] == 3)] = 4; a[] = 0; Table[ a[n], {n, 0, 100}] (* _Jean-François Alcover, Jul 30 2013, after Franklin T. Adams-Watters *)

a(n) = 4 * A055029(n). - Franklin T. Adams-Watters, May 05 2006

More terms from Reiner Martin, Jul 20 2001

A121307 Products of three primes of the form 3n-1 (A003627), not necessarily distinct.

Original entry on oeis.org

8, 20, 44, 50, 68, 92, 110, 116, 125, 164, 170, 188, 212, 230, 236, 242, 275, 284, 290, 332, 356, 374, 404, 410, 425, 428, 452, 470, 506, 524, 530, 548, 575, 578, 590, 596, 605, 638, 668, 692, 710, 716, 725, 764, 782, 788, 830, 890, 902, 908, 932, 935, 956

Offset: 1

Jonathan Vos Post, Sep 05 2006

It would be incorrect to call these Eisenstein 3-almost primes. For the Eisenstein primes see A055664. - N. J. A. Sloane, Feb 06 2008.

J. H. Conway and R. K. Guy, The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
Stan Wagon, "Eisenstein Primes," Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A003627, A112770.
Intersection of A004612 and A014612.
Subsequence of A373589, which in turn is a subsequence of A373597.
Cf. also A055664.

ok[n_] := Block[{f = FactorInteger@n}, Plus @@ Last /@ f == 3 && Max@ Mod[1 + First /@ f, 3] == 0]; Select[Range@ 1000, ok] (* Giovanni Resta, Jun 12 2016 *)

list(lim)=my(v=List(),u=v,t); forprime(p=2,lim\4, if(p%3==2, listput(u,p))); for(i=1,#u, for(j=i,#u, if(u[i]*u[j]^2>lim, break); for(k=j,#u, t=u[i]*u[j]*u[k]; if(t>lim, break); listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

from sympy import primerange
from itertools import combinations_with_replacement as mc
def aupto(limit):
    terms = [p for p in primerange(2, limit//4+1) if p%3 == 2]
    return sorted(set(a*b*c for a, b, c in mc(terms, 3) if a*b*c <= limit))
print(aupto(957)) # Michael S. Branicky, Aug 20 2021

Definition corrected by N. J. A. Sloane, Feb 06 2008
a(37)-a(53) from Giovanni Resta, Jun 12 2016
Name edited by Antti Karttunen, Jun 13 2024

A364869 Numbers k such that 6*k+1 is the norm of an Eisenstein prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 20, 21, 23, 25, 26, 27, 30, 32, 33, 35, 37, 38, 40, 45, 46, 47, 48, 51, 52, 55, 56, 58, 61, 62, 63, 66, 68, 70, 72, 73, 76, 77, 81, 83, 87, 88, 90, 91, 95, 96, 100, 101, 102, 103, 105, 107, 110, 112, 115, 118, 121, 122, 123

Offset: 1

Jianing Song, Aug 11 2023

Numbers k such that 6*k+1 is a prime or the square of a prime congruent to 5 modulo 6.

If p is an Eisenstein prime of norm 6*a(n)+1 (there are two up to association if a(n) is a prime, one if a(n) is the square of a prime), then for any Eisenstein integer x, we have x^a(n) == 0, 1, w, w^2, -1, -w or -w^2 (mod p), where w = (1+sqrt(-3))/2 is a primitive sixth root of unity.

			4 is a term since 6*4+1 is the norm of the Eisenstein prime 5.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A055664, A364868.

Contains 4*A024702 as a subsequence.

isA364869(n) = isA055664(6*n+1) \\ See A055664 for its program

a(n) = (A055664(n+2) - 1)/6.

A112770 Products of pairs of terms from A003627.

Original entry on oeis.org

4, 10, 22, 25, 34, 46, 55, 58, 82, 85, 94, 106, 115, 118, 121, 142, 145, 166, 178, 187, 202, 205, 214, 226, 235, 253, 262, 265, 274, 289, 295, 298, 319, 334, 346, 355, 358, 382, 391, 394, 415, 445, 451, 454, 466, 478, 493, 502, 505, 514, 517, 526, 529, 535

Offset: 1

Jonathan Vos Post and Ray Chandler, Oct 15 2005

It would be incorrect to call these Eisenstein semiprimes. For the Eisenstein primes see A055664. - N. J. A. Sloane, Feb 06 2008. For

Conway, J. H. and Guy, R. K., The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

Eric Weisstein's World of Mathematics, Eisenstein Integer.
Eric Weisstein's World of Mathematics, Eisenstein Prime.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A003627.

{a(n)} = {p*q: p and q both elements of A007645} = {p*q: p and q both of form 3*m^2 * n^2 for integers m, n}.

Definition corrected by N. J. A. Sloane, Feb 06 2008

A120124 Smallest prime p such that p*10^n + 1 is a prime.

Original entry on oeis.org

3, 7, 3, 7, 7, 61, 3, 7, 7, 3, 19, 37, 109, 79, 97, 13, 37, 19, 73, 103, 97, 283, 157, 61, 19, 61, 1213, 3, 163, 691, 367, 163, 181, 157, 241, 3, 103, 733, 151, 283, 337, 193, 211, 163, 7, 73, 307, 61, 223, 1549, 31, 127, 13, 547, 103, 151, 193, 811, 337, 19, 1021, 151

Offset: 1

Alexander Adamchuk, Aug 15 2006

nonn

All terms belong to A007645. All terms also belong to A055664. Also many terms including the first 14 smallest primes from 3 to 139 {3,7,13,19,31,37,43,61,73,79,97,103,127,139} belong tpA023203. The smallest term that differs from A023203 is 151.

			a(1) = 3 because 31 = 3*10 + 1 is the smallest prime of form p*10 + 1, where p is a prime.
a(2) = 7 because 701 = 7*100 + 1 is the smallest prime of form p*100 + 1.

Robert Israel, Table of n, a(n) for n = 1..1000 (first 300 terms from _Vincenzo Librandi_)

Cf. A121172, A030430, A062800, A007645, A055664, A023203.

Primes:= select(isprime,[$1..10^5]):
for n from 1 to 1000 do
   for p in Primes do
      if isprime(p*10^n+1) then
        A[n]:= p
      fi
    od
od:
seq(A[n],n=1..1000); # Robert Israel, May 29 2014

prs=Prime[Range[2000]];Table[i=1;While[!PrimeQ[First[Take[prs,{i}]] 10^n+1],i++];Prime[i],{n,200}] (* Harvey P. Dale, May 15 2011 *)

A134324 Number of Eisenstein-Jacobi primes whose modulus is > n and <= n+1.

Original entry on oeis.org

0, 12, 12, 12, 18, 12, 24, 12, 36, 12, 30, 24, 36, 24, 36, 24, 42, 24, 36, 48, 48, 24, 42, 36, 60, 48, 36, 60, 54, 48, 36, 60, 72, 60, 36, 60, 48, 48, 72, 72, 78, 84, 60, 60, 72, 60, 78, 84, 84, 36, 72, 84, 114, 48

Offset: 0

Philippe Lallouet (philip.lallouet(AT)orange.fr), Jan 30 2008, Feb 06 2008

Cf. A055664, A055665, A055666, A055667, A055668.

a(n) = Sum_{k=n^2+1..(n+1)^2} A055667(k). - Rémy Sigrist, Aug 08 2018

Data corrected and name clarified by Rémy Sigrist, Aug 08 2018

cp's OEIS Frontend

A055028 Number of Gaussian primes of norm n.

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A121307 Products of three primes of the form 3n-1 (A003627), not necessarily distinct.

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A364869 Numbers k such that 6*k+1 is the norm of an Eisenstein prime.

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A112770 Products of pairs of terms from A003627.

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A120124 Smallest prime p such that p*10^n + 1 is a prime.

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Programs

Maple

Mathematica

A134324 Number of Eisenstein-Jacobi primes whose modulus is > n and <= n+1.

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Author

Keywords

Crossrefs

Formula

Extensions