A007645 -id:A007645 - cp's OEIS frontend

A179891 Composites h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is not integer.

4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 56, 57, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 111

Offset: 1

Jaroslav Krizek, Jul 30 2010

nonn

Composites h such that B(h) = A053818(h) / A023896(h) = A175505(h) / A175506(h) is not integer.
Composites h such that A175506(h) > 1.
Subsequence of A179872.
A179872 is the union of this sequence and A007645.

			a(6) = 14 because B(14) = A053818(14) / A023896(14) = 406/42 = 29/3 (not integer).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1749 from G. C. Greubel)

Cf. A007645, A023896, A053818, A175505, A175506.

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890.

f[n_] := 2 Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2)/(n*EulerPhi@n); Select[ Range@ 111, ! PrimeQ@# && ! IntegerQ@f@# &] (* Robert G. Wilson v, Aug 02 2010 *)

isok(k) = if(isprime(k), 0, my(f = factor(k)); if(k == 1, 0, denominator(2*k/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)) > 1)); \\ Amiram Eldar, May 25 2025

More terms from Robert G. Wilson v, Aug 02 2010

A179872 Numbers h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is not integer.

Original entry on oeis.org

3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 48, 49, 50, 51, 52, 54, 56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 86, 87, 88

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Numbers h such that B(h) = A053818(h) / A023896(h) = A175505(h) / A175506(h) is not integer.
Numbers h such that A175506(h) > 1.
Complement of A179871.
Union of A007645 and A179891.

			a(6) = 9 because B(9) = A053818(9) / A023896(9) = 159/27 = 53/9 (not integer).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A179871, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

isok(k) = {my(f = factor(k)); if(k == 1, 0, denominator(2*k/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)) > 1);} \\ Amiram Eldar, May 25 2025

More terms from Amiram Eldar, May 25 2025

A024614 Numbers of the form x^2 + xy + y^2, where x and y are positive integers.

Original entry on oeis.org

3, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199, 201, 208, 211, 217, 219, 223, 228

Offset: 1

Clark Kimberling

Equivalently, sequence A024612 with duplicates removed; i.e., numbers of the form i^2 - i*j + j^2, where 1 <= i < j.
A subsequence of A135412, which consists of multiples (by squarefree factors) of the numbers listed here. It appears that this lists numbers > 1 which have in their factorization: (a) no even power of 3 unless there is a factor == 1 (mod 6); (b) no odd power of 2 or of a prime == 5 (mod 6) and no even power unless there is a factor 3 or == 1 (mod 6). - M. F. Hasler, Aug 17 2016
If we regroup the entries in a triangle with row lengths A004526
   3,
   7,
  12, 13,
  19, 21,
  27, 28, 31,
  37, 39, 43,
... it seems that the j-th row of the triangle contains the numbers i^2+j^2-i*j in row j>=2 and column i = floor((j+1)/2) .. j-1. - R. J. Mathar, Aug 21 2016
Proof of the above characterization: the sequence is the union of 3*(the squares A000290) and A024606 (numbers x^2+xy+y^2 with x > y > 0). For the latter it is known that these are the numbers with a factor p==1 (mod 6) and any prime factor == 2 (mod 3) occurring to an even power. The former (3*n^2) are the same as (odd power of 3)*(even power of any other prime factor). The union of the two cases yields the earlier characterization. - M. F. Hasler, Mar 04 2018
Least term that can be written in exactly n ways is A300419(n). - Altug Alkan, Mar 04 2018
For the general theory see the Fouvry et al. reference and A296095. Bounds used in the Julia program are based on the theorems in this paper. - Peter Luschny, Mar 10 2018

			3 = 1^2 + 1^2 + 1*1, 7 = 2^2 + 1^2 + 2*1, ...

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Peter Luschny)
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Cf. A003136, A007645 (prime terms), A024612, A135412, A296095, A300419.

function isA024614(n)
    n % 3 >= 2 && return false
    n == 3 && return true
    M = Int(round(2*sqrt(n/3)))
    for y in 2:M, x in 1:y
        n == x^2 + y^2 + x*y && return true
    end
    return false
end
A024614list(upto) = [n for n in 1:upto if isA024614(n)]
println(A024614list(228)) # Peter Luschny, Mar 02 2018 updated Mar 17 2018

isA024614 := proc(n)
    local i,j,disc;
    # n=i^2+j^2-i*j = (j-i)^2+i*j, 1<=i=1 and i*j>=j and i^2+j^2-i*j >= 1+j max search radius
    for j from 2 to n-1 do
        # i=(j +- sqrt(4n-3j^2))/2
        disc := 4*n-3*j^2 ;
        if disc >= 0 then
            if issqr(disc) then
                i := (j+sqrt(disc))/2 ;
                if type(i,'integer') and i >= 1 and i= 1 and iA024614(t) then
        printf("%d %d\n",n,t) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 21 2016
# second Maple program:
a:= proc(n) option remember; local k, x;
      for k from a(n-1)+1 do for x while x^2 x^2+(x+y)*y=k)((isqrt(4*k-3*x^2)-x)/2) then return k fi
      od od
    end: a(0):=0:
seq(a(n), n=1..200);  # Alois P. Heinz, Mar 02 2018

max = 228; T0 = {}; xm = Ceiling[Sqrt[max]]; While[T = T0; T0 = Table[x^2 + x y + y^2, {x, 1, xm}, {y, x, xm}] // Flatten // Union // Select[#, # <= max&]&; T != T0, xm = 2 xm]; T (* Jean-François Alcover, Mar 23 2018 *)

is(n)={n>2&&!for(i=1, #n=Set(Col(factor(n)%6))/*consider prime factors mod 6*/, n[i][1]>1||next/*skip factors = 1 mod 6*/; /* odd power: ok only if p=3 */n[i][2]%2&&(n[i][1]!=3 || next) && return; /*even power: ok if there's a p==1, listed first*/ n[1][1]==1 || /*also ok if it's not a 3 and if there's a 3 elsewhere */ (n[i][1]==2 && i<#n && n[i+1][1]==3) || (n[i][1]>3 && for(j=1,i-1,n[j][1]==3 && next(2))||return))} \\ M. F. Hasler, Aug 17 2016, documented & bug fixed (following an observation by Altug Alkan) Mar 04 2018

is(n)={(n=factor(n))||return/*n=1*/; /*exponents % 2, primes % 3*/ n[,2]%=2; n[,1]%=3; (n=Set(Col(n))) /*odd power of a prime == 2? will be last*/ [#n][2] && n[#n][1]==2 && return; /*factor == 1? will be 1st or after 3*/ n[1+(!n[1][1] && #n>1)][1]==1 || /*thrice a square?*/ (!n[1][1]&&n[1][2]&&!for(i=2,#n,n[i][2]&&return))} \\ Alternate code, 5% slower, maybe a bit less obscure. - M. F. Hasler, Mar 04 2018

N=228; v=vector(N);
for(x=1,N, x2=x*x; if(x2>N,break); for(y=x,N, t=x2+y*(x+y); if(t>N,break); v[t]=1;));
for(n=1,N,if(v[n],print1(n,", "))); \\ Joerg Arndt, Mar 10 2018

list(lim)=my(v=List(), x2); lim\=1; for(x=1, sqrtint(4*lim\3), x2=x^2; for(y=1, min((sqrt(4*lim-3*x2)-x)/2, x), listput(v, y*(x+y)+x2))); Set(v) \\ Charles R Greathouse IV, Mar 23 2018

Edited by M. F. Hasler, Aug 17 2016

b-file for values a(1)..a(10^4) double-checked with PARI code by M. F. Hasler, Mar 04 2018

A034017 Numbers that are primitively represented by x^2 + xy + y^2.

Original entry on oeis.org

0, 1, 3, 7, 13, 19, 21, 31, 37, 39, 43, 49, 57, 61, 67, 73, 79, 91, 93, 97, 103, 109, 111, 127, 129, 133, 139, 147, 151, 157, 163, 169, 181, 183, 193, 199, 201, 211, 217, 219, 223, 229, 237, 241, 247, 259, 271, 273, 277, 283, 291, 301, 307, 309, 313, 327, 331

Offset: 1

N. J. A. Sloane

Gives the location of the nonzero terms of A000086.
Starting at a(3), a(n)^2 is the ordered semiperimeter of primitive integer Soddyian triangles (see A210484). - Frank M Jackson, Feb 04 2013
A000086(a(n)) > 0; a(n) = A004611(k) or a(n) = 3*A004611(k) for n > 3 and an appropriate k. - Reinhard Zumkeller, Jun 23 2013
The number of structure units in an icosahedral virus is 20*a(n), see Stannard link. - Charles R Greathouse IV, Nov 03 2015
From Wolfdieter Lang, Apr 09 2021: (Start)
The positive definite binary quadratic form F = [1, 1, 1], that is x^2 + x*y + y^2, has discriminant Disc = -3, and class number 1 (see Buell, Examples, p. 19, first line: Delta = -3, h = 1). This reduced form is equivalent to the form [1,-1, 1], but to no other reduced one (see Buell, Theorem 2.4, p. 15).
This form F represents a positive integer k (= a(n)) properly if and only if A002061(j+1) = 2*T(j) + 1 = j^2 + j + 1 == 0 (mod k), for j from {0, 1, ..., k-1}. This congruence determines the representative parallel primitive forms (rpapfs) of discriminant Disc = -3 and representation of a positive integer number k, given by [k, 2*j+1, c(j)], and c(j) is  determined from Disc =-3 as c(j) = ((2*j+1)^2 + 3)/(4*k) = (j^2 + j + 1)/k. Each rpapf has a first reduced form, the so-called right neighbor form, namely [1, 1, 1] for k = 1 = a(1) (the already reduced parallel form from j = 0), and [1, -1, 1] for k = a(n), with n >= 2.
Only odd numbers k are eligible for representation, because 2*T(j) + 1, with the triangular numbers T = A000217, is odd. The odd k with at least one solution of the congruence are then the members of the present sequence.
The solutions of the reduced forms F = [1, 1, 1] and F' = [1, -1, 1] representing k are related by type I equivalence because of the first two entries ([a, a, c] == [a, -a, c]), and also by type II equivalence because [a, b, a] == [a, -b, a], for positive b. These transformation matrices are R_I =  Matrix([1, -1],[0, 1]) and R_{II} = Matrix([0, -1], [1, 0]), respectively, to obtain the forms with negative second entry from the ones with positive second entry. The corresponding solutions (x, y)^t (t for transposed) are related by the inverse of these matrices.
The table with the A341422(n) solutions j of the congruence given above are given in A343232. (End)
Apparently, also the integers k that can be expressed as a quotient of two terms from A002061. - Martin Becker, Aug 14 2022
For some x, y let a(n) = r, x*(x+y) = s, y*(x+y) = t, x*y = u then (r,s,t,u) is a Pythagorean quadruple such that r^2 = s^2 + t^2 + u^2. - Frank M Jackson, Feb 26 2024

B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 184, AMS, Providence, RI, 1995.
D. A. Buell, Binary Quadratic Forms, Springer, 1989, pp. 15, 19.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Linda Stannard, Principles of Virus Architecture (1995).

Cf. A000217, A002061, A002476, A003136, A007645 (primes), A045611, A045897, A226946 (complement), A045897 (subsequence), A341422, A343232.

a034017 n = a034017_list !! (n-1)
a034017_list = 0 : filter ((> 0) . a000086) [1..]
-- Reinhard Zumkeller, Jun 23 2013

N:= 1000: # to get all terms <= N
P:= select(isprime, [seq(6*n+1, n=1..floor((N-1)/6))]):
A:= {1,3}:
for p in P do
  A:= {seq(seq(a*p^k, k=0..floor(log[p](N/a))),a=A)}:
od:
sort(convert(A,list)); # Robert Israel, Nov 04 2015

lst = {0}; maxLen = 331; Do[If[Reduce[m^2 + m*n + n^2 == k && m >= n >= 0 && GCD[m, n] == 1, {m, n}, Integers] === False, , AppendTo[lst, k]], {k, maxLen}]; lst (* Frank M Jackson, Jan 10 2013 *) (* simplified by T. D. Noe, Feb 05 2013 *)

is(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,1]%3!=1 && (f[i,1]!=3 || f[i,2]>1), return(n==0))); 1 \\ Charles R Greathouse IV, Jan 10 2013

list(lim)=if(lim<7, return(select(n->n<=lim, [0,1,3]))); my(v=List([0,1,3])); for(x=1,sqrtint(lim\=1), my(y,t); while(y++Charles R Greathouse IV, Jan 20 2022

n >= 2: 3^{0 or 1} X product of primes of form 3a+1 (A002476) to any nonnegative power.

The sequence {a(n)}{n>=2} gives the increasingly sorted positive numbers k such that the set M(k) := {j = 0, 1, 2, ..., k-1 | j^2 + j + 1 == 0 (mod k)}, has cardinality >= 1. - _Wolfdieter Lang, Apr 09 2021

Extended by Ray Chandler, Jan 29 2009

A026225 Numbers of the form 3^i * (3k+1).

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 30, 31, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 55, 57, 58, 61, 63, 64, 66, 67, 70, 73, 75, 76, 79, 81, 82, 84, 85, 88, 90, 91, 93, 94, 97, 100, 102, 103, 106, 108, 109, 111, 112, 115

Offset: 1

Clark Kimberling

Old name: a(n) = (1/3)*(s(n+1) - 1), where s = A026224.
Conjectures based on old name: these are numbers of the form (3*i+1)*3^j; see A182828, and they comprise the complement of A026179, except for the initial 1 in A026179.
From Peter Munn, Mar 17 2022: (Start)
Numbers with an even number of prime factors of the form 3k-1 counting repetitions.
Numbers whose squarefree part is congruent to 1 modulo 3 or 3 modulo 9.
The integers in an index 2 subgroup of the positive rationals under multiplication. As such the sequence is closed under multiplication and - where the result is an integer - under division; also for any positive integer k not in the sequence, the sequence's complement is generated by dividing by k the terms that are multiples of k.
Alternatively, the sequence can be viewed as an index 2 subgroup of the positive integers under the commutative binary operation A059897(.,.).
Viewed either way, the sequence corresponds to a subgroup of the quotient group derived in the corresponding way from A055047. (End)
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022
Is this A026140 shifted right? - R. J. Mathar, Jun 24 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree Part.

Elements of array A182828 in ascending order.
Union of A055041 and A055047.
Other subsequences: A007645 (primes), A352274.
Symmetric difference of A003159 and A225838; of A007417 and A189716.
Cf. A001222, A059897, A343430.

a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, 160}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[3, 1]  (* A026225 *)
p[3, 2] (* A026179 without initial 1 *)
(* Clark Kimberling, Oct 19 2016 *)

isok(m) = core(m) % 3 == 1 || core(m) % 9 == 3; \\ Peter Munn, Mar 17 2022

from sympy import integer_log
def A026225(n):
    def f(x): return n+x-sum(((x//3**i)-1)//3+1 for i in range(integer_log(x,3)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 15 2025

From Peter Munn, Mar 17 2022: (Start)
{a(n) : n >= 1} = {m : A001222(A343430(m)) == 0 (mod 2)}.
{a(n) : n >= 1} = {A055047(m) : m >= 1} U {3*A055047(m) : m >= 1}.
{a(n) : n >= 1} = {A352274(m) : m >= 1} U {A352274(m)/10 : m >= 1, 10 divides A352274(m)}. (End)

New name from Peter Munn, Mar 17 2022

A066498 Numbers k such that 3 divides phi(k).

Original entry on oeis.org

7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 31, 35, 36, 37, 38, 39, 42, 43, 45, 49, 52, 54, 56, 57, 61, 62, 63, 65, 67, 70, 72, 73, 74, 76, 77, 78, 79, 81, 84, 86, 90, 91, 93, 95, 97, 98, 99, 103, 104, 105, 108, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 133

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Numbers k such that x^3 == 1 (mod k) has solutions 1 < x < k.
Terms are multiple of 9 or of a prime of the form 6k+1.
If k is a term of this sequence, then G =  is a non-abelian group of order 3k, where 1 < r < n and r^3 == 1 (mod k). For example, G can be the subgroup of GL(2, Z_k) generated by x = {{1, 1}, {0, 1}} and y = {{r, 0}, {0, 1}}. - Jianing Song, Sep 17 2019
The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, Mar 21 2021

			If n < 7 then x^3 = 1 (mod n) has no solution 1 < x < n, but {2,4} are solutions to x^3 == 1 (mod 7), hence a(1) = 7.

Harry J. Smith, Table of n, a(n) for n=1..1000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Complement of A088232.
Cf. A000010, A066499, A066500, A066501, A066502.
A007645 gives the primes congruent to 1 mod 3.
Column k=2 of A277915.

Select[Range[150], Divisible[EulerPhi[#], 3]&] (* Harvey P. Dale, Jan 12 2011 *)

isok(k)={ eulerphi(k)%3 == 0 } \\ Harry J. Smith, Feb 18 2010

Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003

Corrected and extended by Ray Chandler, Nov 05 2003

A345435 Represent the ring of Eisenstein integers E = {x+yomega: x, y rational integers, omega = exp(2Pi*i/3)} by the cells of a hexagonal grid; number the cells of the grid along a counterclockwise hexagonal spiral, with the cells 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Original entry on oeis.org

0, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 91, 95, 97, 101, 103, 107, 109, 113, 115

Offset: 1

N. J. A. Sloane, Jun 23 2021

nonn

The Eisenstein integer represented by cell m is A307013(m) + A307012(m)*omega. Thus the set of Eisenstein primes is {A307013(a(n)) + A307012(a(n))*omega : n >= 2}. - Peter Munn, Jun 26 2021
The Eisenstein integer a + b*omega has norm a^2 - a*b + b^2 (see A003136). The number of Eisenstein integers of norm n is given by A004016(n).
The norms of the Eisenstein primes are given in A055664, and the number of Eisenstein primes of norm n is given in A055667.
Reid's 1910 book (still in print) is still the best reference for the Eisenstein integers and similar rings.

			The smallest Eisenstein integers are 0 (of norm 0), and the six units of norm 1, namely (writing w for omega) +-1, +-w, +-w^2.
The first few Eisenstein primes are (here u is any of the six units):
   u*(2+w), norm = 3, number = 6;
   2*u, norm = 4, number = 6;
   u*(3+w), norm = 7, number = 6;
   u*(3+2*w), norm = 7, number = 6 (so there are 12 primes of norm 7 - see A055667).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.4, p. 111.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A345435
N. J. A. Sloane, Illustration of initial terms [An enlargement of Figure 1 of Wichmann (2019), showing the numbering of the initial cells of the hexagonal spiral.]
Eric Weisstein's World of Mathematics, Eisenstein Prime
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019.
Brian Wichmann, The Eisenstein integers, with the primes shaded [Figure 1 from the previous link]

Cf. A003136, A003173, A003627, A004016, A007645, A055664, A055667, A307012, A307013, A308412, A345436, A345437.

PARI
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See Links section.
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More terms from Rémy Sigrist, Jun 26 2021

A045331 Primes congruent to {1, 2, 3} mod 6; or, -3 is a square mod p.

Original entry on oeis.org

2, 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

-3 is a quadratic residue mod a prime p iff p is in this sequence.

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

Apart from initial term, same as A007645; apart from initial two terms, same as A002476.
Cf. A002313, A033203, A038873, A038874, A057128.
Subsequence of A047246.

a045331 n = a045331_list !! (n-1)
a045331_list = filter ((< 4) . (`mod` 6)) a000040_list
-- Reinhard Zumkeller, Jan 15 2013

[p: p in PrimesUpTo(700) | p mod 6 in [1, 2, 3]]; // Vincenzo Librandi, Aug 08 2012

Select[Prime[Range[200]],MemberQ[{1,2,3},Mod[#,6]]&]  (* Harvey P. Dale, Mar 31 2011 *)
Join[{2,3},Select[Range[7,10^3,6],PrimeQ]] (* Zak Seidov, May 20 2011 *)

select(n->n%6<5,primes(100)) \\ Charles R Greathouse IV, May 20 2011

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

A002648 A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2.

Original entry on oeis.org

13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249, 129793, 139969, 142573, 147853, 169933

Offset: 1

N. J. A. Sloane

nonn

Primes p such that p = 1 + 3*m^2 for some integer m (A111051). - Michael Somos, Sep 15 2005

			193 is a term since 193 = (9^3 - 7^3)/(9 - 7) is a prime.

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
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).

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Eric Weisstein's World of Mathematics, Cuban Prime.
Wikipedia, Cuban prime.

Cf. A002407, A111051 (values of m).

A subsequence of A007645.

[a: n in [0..400] | IsPrime(a) where a is 3*n^2+1]; // Vincenzo Librandi, Dec 02 2011

Select[Table[3n^2+1,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)

{a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( cMichael Somos, Sep 15 2005 */

a(n) = 3*A111051(n)^2 + 1. - Paul F. Marrero Romero, Nov 03 2023

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

cp's OEIS Frontend

A179891 Composites h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is not integer.

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A179872 Numbers h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is not integer.

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A024614 Numbers of the form x^2 + xy + y^2, where x and y are positive integers.

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A034017 Numbers that are primitively represented by x^2 + xy + y^2.

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A026225 Numbers of the form 3^i * (3k+1).

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A066498 Numbers k such that 3 divides phi(k).

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