A007645 -id:A007645 - cp's OEIS frontend

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

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 *)

A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

Original entry on oeis.org

3, 4, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 80, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199

Offset: 1

Peter Luschny, Mar 03 2018

nonn

Equivalently these are the integers represented by a cyclotomic binary form Phi_p(x,y) where p is prime and x and y are positive integers with max(x,y) >= 2. A cyclotomic binary form (over Z) is a homogeneous polynomial in two variables of the form f(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function.

An efficient and safe calculation of this sequence requires a precise knowledge of the range of possible solutions of the associated Diophantine equations. The bounds used in the Julia program below were specified by Fouvry, Levesque and Waldschmidt.

			Let p denote an odd prime. Subsequences are numbers of the form
2^p - 1,         (A001348) (x = 1, y = 2) (Mersenne numbers),
p*2^(p - 1),     (A299795) (x = 2, y = 2),
(3^p - 1)/2,     (A003462) (x = 1, y = 3),
3^p - 2^p,       (A135171) (x = 2, y = 3),
p*3^(p - 1),     (A027471) (x = 3, y = 3),
(4^p - 1)/3,     (A002450) (x = 1, y = 4),
2^(p-1)*(2^p-1), (A006516) (x = 2, y = 4),
4^p - 3^p,       (A005061) (x = 3, y = 4),
p*4^(p - 1),     (A002697) (x = 4, y = 4),
(p^p-1)/(p-1),   (A023037),
p^p,             (A000312, A051674).
.
The generalized cuban primes A007645 are a subsequence, as are the quintan primes A002649, the septan primes and so on.
All primes in this sequence less than 1031 are generalized cuban primes. 1031 is an element because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, however 1031 is not a cuban prime because 1030 is not divisible by 6.

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

Indices of the nonzero values of A300333.

Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645.

using Primes
function isA300332(n)
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 2
    while k <= K
        if k == 7
            K = Int(floor(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            r = x == y ? k*y^(k - 1) : div(x^k - y^k, x - y)
            n == r && return true
        end
        k = nextprime(k+1)
    end
    return false
end
A300332list(upto) = [n for n in 1:upto if isA300332(n)]
println(A300332list(200))

A354092 Fully multiplicative prime shift where the primes of the form 3k+2 are replaced by the previous such prime (with 2 -> 1), and primes of the form 3k and 3k+1 stay as they are.

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 7, 1, 9, 2, 5, 3, 13, 7, 6, 1, 11, 9, 19, 2, 21, 5, 17, 3, 4, 13, 27, 7, 23, 6, 31, 1, 15, 11, 14, 9, 37, 19, 39, 2, 29, 21, 43, 5, 18, 17, 41, 3, 49, 4, 33, 13, 47, 27, 10, 7, 57, 23, 53, 6, 61, 31, 63, 1, 26, 15, 67, 11, 51, 14, 59, 9, 73, 37, 12, 19, 35, 39, 79, 2, 81, 29, 71, 21, 22, 43, 69, 5, 83

Offset: 1

Antti Karttunen, May 17 2022

Index entries for sequences computed from indices in prime factorization

Left inverse of A354091.
Cf. A003627, A007645.
Cf. A064989, A348747 (variants).

A354092(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), if(2==f[k,1], f[k,1]--, forstep(i=primepi(f[k,1])-1,0,-1,if(2==(prime(i)%3), f[k,1]=prime(i); break))))); factorback(f); };

Fully multiplicative with a(2) = 1, a(A003627(1+n)) = A003627(n), a(A007645(n)) = A007645(n).

For all n >= 1, a(A354091(n)) = n.

A034934 Numbers k such that (3*k + 1)/2 is prime.

Original entry on oeis.org

1, 3, 7, 11, 15, 19, 27, 31, 35, 39, 47, 55, 59, 67, 71, 75, 87, 91, 99, 111, 115, 119, 127, 131, 151, 155, 159, 167, 171, 175, 179, 187, 195, 207, 211, 231, 235, 239, 255, 259, 267, 279, 287, 295, 299, 307, 311, 319, 327, 335, 339, 347, 371, 375, 379, 391

Offset: 1

Jud McCranie

nonn

Related to hyperperfect numbers of a certain form.

The formula by Jaroslav Krizek is explained as follows: If p = (3n+1)/2 is prime, then it is an integer, and p must be of the form p = 3m-1, i.e., p = A003627(k). On the other hand, if p = A003627(k), then all k < p are coprime to p, so we have B(p) = (Sum_{kM. F. Hasler, Nov 29 2010

			a(6) = 19 because for A003627(6) = 29, B(29) = A053818(29)/A023896(29) = 7714/406 = 19. Cf. A179871-A179891, A003627, A007645. - _Jaroslav Krizek_, Aug 01 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000
Judson S. McCranie, A study of hyperperfect numbers, J. Int. Seq., Vol. 3 (2000), Article 00.1.3.

Cf. A003627, A007645, A038536, A045309, A175505, A179871-A179891.

[ n: n in [1..400 by 2] | IsPrime((3*n+1) div 2) ];

Select[Range[500], PrimeQ[(3# + 1)/2] &] (* Harvey P. Dale, Jan 15 2011 *)

is(n)=isprime((3*n+1)/2) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A175505(A003627(n)). - Jaroslav Krizek, Aug 01 2010

Corrected by Vincenzo Librandi, Mar 24 2010

A139494 Primes of the form x^2 + 11x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

13, 43, 61, 79, 103, 127, 139, 157, 181, 199, 211, 277, 283, 313, 337, 367, 373, 433, 439, 523, 547, 571, 601, 607, 673, 727, 751, 757, 823, 829, 859, 883, 907, 919, 937, 991, 997, 1039, 1063, 1069, 1093, 1117, 1153, 1171, 1213, 1231, 1249, 1291, 1297

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 11; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

193, 337, 457, 673, 1009, 1033, 1129, 1201, 1297, 1801, 1873, 2017, 2137, 2377, 2473, 2521, 2689, 2713, 2857, 3049, 3217, 3313, 3361, 3529, 3697, 3889, 4057, 4153, 4201, 4561, 4657, 4729, 4993, 5209, 5233, 5569, 5737, 5881, 6073, 6217, 6337, 6553, 6577

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Also primes of the form x^2 + 168y^2. - T. D. Noe, Apr 29 2008

In base 12, the sequence is 141, 241, 321, 481, 701, 721, 7X1, 841, 901, 1061, 1101, 1201, 12X1, 1461, 1521, 1561, 1681, 16X1, 17X1, 1921, 1X41, 1E01, 1E41, 2061, 2181, 2301, 2421, 24X1, 2521, 2781, 2841, 28X1, 2X81, 3021, 3041, 3281, 33X1, 34X1, 3621, 3721, 3801, 3961, 3981, where X is 10 and E is 11. Moreover, the discriminant is 480. - Walter Kehowski, Jun 01 2008

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

a = {}; w = 26; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

The primes are congruent to {1, 25, 121} (mod 168). - T. D. Noe, Apr 29 2008

A139512 Primes of the form x^2 + 32xy + y^2 for x and y nonnegative.

Original entry on oeis.org

229, 349, 409, 421, 661, 769, 829, 1021, 1069, 1249, 1381, 1429, 1549, 1789, 1801, 1861, 2089, 2161, 2269, 2389, 3001, 3061, 3109, 3181, 3229, 3469, 3889, 4021, 4129, 4201, 4441, 4861, 4909, 5101, 5449, 5521, 5869, 5881, 6121, 6469, 6481, 6529, 6781

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Are all terms == 1 mod 12? - Zak Seidov, Apr 25 2008

Yes: (i) all terms == 1 mod 3 because the quadratic form has terms == {0,1} mod 3 and the values ==0 mod 3 are not primes. (ii) all terms == 1 mod 4 because the quadratic form has terms == {0,1,2} mod 4 and the values = {0,2} mod 4 are not primes. By the Chinese remainder constructions for coprime 3 and 4 all prime terms are == 1 mod 12. - R. J. Mathar, Jun 10 2020

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references), discriminant 1020.

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 32; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A171715 Absolute value of (n-th prime of form 3m-1 minus n-th prime of form 3k+1/2+-1/2).

Original entry on oeis.org

1, 2, 2, 2, 8, 8, 2, 14, 14, 14, 8, 14, 14, 8, 20, 26, 20, 20, 14, 14, 20, 20, 20, 26, 2, 8, 32, 26, 26, 44, 44, 50, 44, 38, 50, 26, 26, 38, 26, 32, 32, 20, 26, 20, 38, 38, 56, 62, 56, 68, 68, 80, 50, 50, 50, 44, 50, 62, 56, 50, 62, 74, 74, 62, 68, 56, 50, 44, 50, 50, 32, 44, 38

Offset: 1

Juri-Stepan Gerasimov, Dec 17 2009, Feb 09 02 2010

nonn

Also, the absolute value of (n-th generalized cuban prime minus n-th generalized non-cuban prime).
Or, the absolute value of n-th prime of form 6*m-3/2+-5/2 minus n-th prime of form 6*k-2+-1.
A003627 U A007645 = A045375 U A045410 = A000040.

			a(1) = abs(3*1-1-(3*1+1/2-1/2)) = 1; a(2) = abs(3*2-1-(3*2+1/2+1/2)) = 2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A003627, A007645, A045375, A045410.

A003627 := proc(n) if n <= 2 then op(n,[2,5]) ; ; else for a from procname(n-1)+2 by 2 do if isprime(a) and (a mod 3) =2 then return a ; end if; end do: end if; end proc:
A007645 := proc(n) if n <= 2 then op(n,[3,7]) ; ; else for a from procname(n-1)+2 by 2 do if isprime(a) and (a mod 3) <> 2 then return a ; end if; end do: end if; end proc:
A171715 := proc(n) abs(A003627(n)-A007645(n)) ; end proc: # R. J. Mathar, Apr 24 2010

Module[{nn=500,p1,p2,len},p1=Select[3*Range[nn]-1,PrimeQ];p2=Select[Flatten[#+{0,1}&/@ (3*Range[nn])],PrimeQ];len=Min[Length[p1],Length[p2]]; Abs[#[[1]]-#[[2]]]&/@ Thread[ {Take[p1,len],Take[p2,len]}]] (* Harvey P. Dale, Aug 29 2023 *)

a(n) = abs(A003627(n)-A007645(n)) = abs(A045375(n)-A045410(n)).

Entries checked by R. J. Mathar, Apr 24 2010

A221717 Non-cuban primes.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 277, 281, 283, 293

Offset: 1

N. J. A. Sloane, Jan 29 2013

nonn

Primes not in A002407.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002407, A007645, A003627.

nn = 10; c = Select[Table[3 x^2 + 3 x + 1, {x, nn}], PrimeQ[#] &]; Complement[Prime[Range[PrimePi[c[[-1]]]]], c] (* T. D. Noe, Jan 30 2013 *)

A139505 Primes of the form x^2 + 25x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

151, 163, 307, 397, 409, 541, 547, 601, 673, 811, 823, 859, 967, 997, 1153, 1231, 1237, 1327, 1567, 1669, 1741, 1879, 2083, 2143, 2281, 2293, 2557, 2677, 2707, 2833, 2971, 3037, 3259, 3313, 3433, 3877, 4003, 4129, 4153, 4603, 4639, 4861, 4957, 5101, 5227

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..5000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 25; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)
With[{nn=80},Select[Union[#[[1]]^2+25#[[1]]#[[2]]+#[[2]]^2&/@Tuples[ Range[ 0,nn],2]],PrimeQ[#]&&#Harvey P. Dale, Feb 10 2020 *)

cp's OEIS Frontend

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

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A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

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Julia

A354092 Fully multiplicative prime shift where the primes of the form 3k+2 are replaced by the previous such prime (with 2 -> 1), and primes of the form 3k and 3k+1 stay as they are.

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A034934 Numbers k such that (3*k + 1)/2 is prime.

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Magma

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Extensions

A139494 Primes of the form x^2 + 11x*y + y^2 for x and y nonnegative.

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Mathematica

A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

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Mathematica

Formula

A139512 Primes of the form x^2 + 32*x*y + y^2 for x and y nonnegative.

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Mathematica

A171715 Absolute value of (n-th prime of form 3*m-1 minus n-th prime of form 3*k+1/2+-1/2).

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A139512 Primes of the form x^2 + 32xy + y^2 for x and y nonnegative.

A171715 Absolute value of (n-th prime of form 3m-1 minus n-th prime of form 3k+1/2+-1/2).