A007519 -id:A007519 - cp's OEIS frontend

A161998 Numbers n such that n^6 + 272 is prime.

2163, 2541, 2667, 4011, 5187, 5733, 5985, 7119, 7371, 7707, 8547, 10017, 10731, 12579, 13041, 13125, 13293, 14007, 14679, 15855, 16317, 16401, 16863, 17283, 19131, 19383, 20139, 20475, 21021, 21357, 22197, 22995, 23457, 23667, 24591, 25053, 25389, 25641

Offset: 1

Arkadiusz Wesolowski, Jun 24 2009

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 272

Cf. A007519, A163592, A066386, A126893, A119276.

[n : n in [1..25641 by 2] | IsPrime(n^6+272)]; // Arkadiusz Wesolowski, Mar 05 2011

Select[Range[1, 25641, 2], PrimeQ[#^6 + 272] &] (* Arkadiusz Wesolowski, Mar 05 2011 *)

forstep(n=1, 25641, 2, if(isprime(n^6+272), print1(n, ", "))); \\ Arkadiusz Wesolowski, Mar 05 2011

A209544 Primes not expressed in form n<+>2, where operation <+> defined in A206853.

Original entry on oeis.org

3, 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1297, 1321

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 10 2012

Trivially every odd prime is expressed in form n<+>1 (cf. A208982).

Are these related to A141174, A045390 or A007519? - R. J. Mathar, Mar 13 2012

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A007519, A209554, A182187.

For n>=2, a(n) = A007519(n-1). - Vladimir Shevelev, Apr 18 2012

A242663 Nonnegative integers of the form x^2 + 4xy - 4*y^2.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 17, 25, 28, 32, 36, 41, 49, 56, 64, 68, 72, 73, 81, 89, 92, 97, 100, 112, 113, 121, 124, 128, 136, 137, 144, 153, 161, 164, 169, 184, 188, 193, 196, 200, 217, 224, 225, 233, 241, 248, 252, 256, 257, 272, 281, 284, 288, 289, 292

Offset: 1

N. J. A. Sloane, May 31 2014

nonn

Discriminant 32.
Also numbers representable as x^2 + 6*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018
Also numbers of the form x^2 - 8*y^2. - Jianing Song, Jul 31 2018

Gheorghe Coserea, Table of n, a(n) for n = 1..100001
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A031363.

Primes in this sequence = A007519.

Reap[For[n = 0, n <= 300, n++, If[Reduce[ x^2 + 4*x*y - 4*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

seq(M, k=6) = {
setintersect([1..M], setbinop((x, y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
concat(0, seq(292)) \\ Gheorghe Coserea, Jul 31 2018

A343104 Smallest number having exactly n divisors of the form 8*k + 1.

Original entry on oeis.org

1, 9, 81, 153, 891, 1377, 8019, 3825, 11025, 15147, 88209, 31977, 354375, 99225, 121275, 95931, 7144929, 187425, 893025, 287793, 1403325, 1499553, 1715175, 675675, 1091475, 6024375, 1576575, 1686825, 72335025, 2027025, 2264802453041139, 2297295, 11609325, 121463793, 9823275

Offset: 1

Jianing Song, Apr 05 2021

nonn

Smallest index of n in A188169.
a(n) exists for all n, since 3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 3^0, 3^2, ..., 3^(2n-2). This actually gives an upper bound for a(n).
From David A. Corneth, Apr 05 2021: (Start)
All terms are odd since if a term is even then the odd part has the same number of such divisors.
No a(2*k + 1) is divisible by a prime congruent to 1 (mod 8).
If for some k, A188169(k) > m then A188169(k*t) > m for all t > 0. This can be used to trim searches when looking for some a(m).
If gcd(k, m) = 1 then A188169(k) * A188169(m) <= A188169(k*m) (End)

			a(4) = 153 since it is the smallest number with exactly 4 divisors congruent to 1 modulo 8, namely 1, 9, 17 and 153.

Smallest number having exactly n divisors of the form 8*k + i: this sequence (i=1), A343105 (i=3), A343106 (i=5), A188226 (i=7).
Cf. A188169.
Cf. A007519, A007520, A007521, A007522.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
a(n) = if(n>0, for(k=1, oo, if(res(k,8,1)==n, return(k))))

a(2n-1) <= 3^(2n-2) * 11, since 3^(2n-2) * 11 has exactly 2n-1 divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^(2n-2), 3^1 * 11, 3^3 * 11, ..., 3^(2n-3) * 11.

a(2n) <= 3^(n-1) * 187, since 3^(n-1) * 187 has exactly 2n divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^b, 3^0 * 17, 3^2 * 17, ..., 3^b * 17, 3^1 * 11, 3^3 * 11, ..., 3^a * 11, 3^1 * 187, 3^3 * 187, ... 3^a * 187, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.

More terms from David A. Corneth, Apr 06 2021

A014755 3 and -3 are both 4th powers (one implies other) mod these primes p=1 mod 8.

Original entry on oeis.org

193, 313, 433, 577, 601, 673, 769, 937, 1201, 1297, 1321, 1657, 1801, 1993, 2137, 2473, 2521, 2593, 2833, 2953, 3169, 3529, 3673, 3697, 3769, 3889, 4057, 4129, 4153, 4297, 4441, 4513, 4561, 4801, 4969, 5113, 5209, 5233, 5281, 5449, 5521

Offset: 1

Warren D. Smith

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007519.

filter:= proc(p) isprime(p) and [msolve(x^4=3, p)] <> [] end proc:
select(filter, [seq(i,i=1..10^4, 8)]); # Robert Israel, May 07 2019

okQ[p_] := PrimeQ[p] && Solve[x^4 == 3, x, Modulus -> p] != {};
Select[Range[1, 10000, 8], okQ] (* Jean-François Alcover, Feb 08 2023 *)

forprime(p=1,9999,p%8==1&&ispower(Mod(3,p),4)&&print1(p",")) \\ M. F. Hasler, Feb 18 2014

is_A014755(p)={p%8==1&&ispower(Mod(3,p),4)&&isprime(p)} \\ M. F. Hasler, Feb 18 2014

from itertools import count, islice
from sympy import nextprime, is_nthpow_residue
def A014755_gen(startvalue=2): # generator of terms >= startvalue
    p = max(nextprime(startvalue-1),2)
    while True:
        if p&7==1 and is_nthpow_residue(3,4,p) and is_nthpow_residue(-3,4,p):
            yield p
        p = nextprime(p)
A014755_list = list(islice(A014755_gen(),20)) # Chai Wah Wu, May 02 2024

Offset changed from 0 to 1 by Bruno Berselli, Feb 20 2014

A045390 Primes congruent to {1, 2} mod 8.

Original entry on oeis.org

2, 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129

Offset: 1

N. J. A. Sloane

Essentially the same as A007519: 2 followed by A007519.

Primes p such that -1 is a 4th power (mod p). E.g.: 1^4 == -1 (mod 2), 2^4 == -1 (mod 17), 3^4 == -1 (mod 41), 10^4 == -1 (mod 73). - Eric M. Schmidt, Mar 27 2014

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A000040.

[p: p in PrimesUpTo(1300) | p mod 8 in [1, 2]]; // Vincenzo Librandi, Aug 11 2012

Select[Prime[Range[300]],MemberQ[{1,2},Mod[#,8]]&] (* Vincenzo Librandi, Aug 11 2012 *)

A060940 Triangle in which n-th row gives the phi(n) terms appearing as initial primes in arithmetic progressions with difference n, with initial term equal to the smallest positive residue coprimes to n.

Original entry on oeis.org

2, 3, 7, 5, 5, 7, 11, 7, 13, 19, 7, 11, 29, 23, 17, 11, 19, 13, 17, 11, 13, 23, 19, 11, 13, 23, 43, 17, 11, 13, 17, 19, 23, 13, 47, 37, 71, 17, 29, 19, 31, 43, 13, 17, 19, 23, 53, 41, 29, 17, 31, 19, 59, 47, 61, 23, 37, 103, 29, 17, 19, 23, 53, 41, 31, 17, 19, 37, 23, 41, 43, 29

Offset: 1

Labos Elemer, May 07 2001

			For differences 1, 2, 3, 4, 5, 6, 7, .. the initial primes are 2; 3; 7, 5; 5, 7; 11, 7, 13, 19; 7, 11; 29, 23, 17, 11, 19, 13; ... etc. Suitable initial values (coprimes to difference) are in A038566. Position of end(start) of rows is given by values of A002088.
From _Seiichi Manyama_, Apr 02 2018: (Start)
   n | phi(n)|
  ---+-------+------------------------
   1 |   1   |  2;
   2 |   1   |  3;
   3 |   2   |  7,  5;
   4 |   2   |  5,  7;
   5 |   4   | 11,  7, 13, 19;
   6 |   2   |  7, 11;
   7 |   6   | 29, 23, 17, 11, 19, 13;
   8 |   4   | 17, 11, 13, 23;
   9 |   6   | 19, 11, 13, 23, 43, 17;
  10 |   4   | 11, 13, 17, 19;         (End)

Seiichi Manyama, Rows n = 1..200, flattened
Eric Weisstein's MathWorld, Dirichlet's theorem
Wikipedia, Dirichlet's theorem on arithmetic progressions

Cf. A000010 (phi), A002088, A038566, A034693, A034694, A002144, A007519, A088732, etc..

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*)

cp's OEIS Frontend

A161998 Numbers n such that n^6 + 272 is prime.

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Magma

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PARI

A209544 Primes not expressed in form n<+>2, where operation <+> defined in A206853.

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A242663 Nonnegative integers of the form x^2 + 4*x*y - 4*y^2.

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Mathematica

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A343104 Smallest number having exactly n divisors of the form 8*k + 1.

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PARI

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Extensions

A014755 3 and -3 are both 4th powers (one implies other) mod these primes p=1 mod 8.

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Maple

Mathematica

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PARI

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Extensions

A045390 Primes congruent to {1, 2} mod 8.

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Magma

Mathematica

A060940 Triangle in which n-th row gives the phi(n) terms appearing as initial primes in arithmetic progressions with difference n, with initial term equal to the smallest positive residue coprimes to n.

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Author

Keywords

Examples

Links

Crossrefs

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

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

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Author

A242663 Nonnegative integers of the form x^2 + 4xy - 4*y^2.

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