A107007 -id:A107007 - cp's OEIS frontend

A140633 Primes of the form 7x^2+4xy+52y^2.

7, 103, 127, 223, 367, 463, 487, 607, 727, 823, 967, 1063, 1087, 1303, 1327, 1423, 1447, 1543, 1567, 1663, 1783, 2143, 2287, 2383, 2503, 2647, 2767, 2887, 3343, 3463, 3583, 3607, 3727, 3823, 3847, 3943, 3967, 4327, 4423, 4447, 4567, 4663

Offset: 1

T. D. Noe, May 19 2008

Discriminant=-1440. Also primes of the forms 7x^2+6xy+87y^2 and 7x^2+2xy+103y^2.
Voight proves that there are exactly 69 equivalence classes of positive definite binary quadratic forms that represent almost the same primes. 48 of those quadratic forms are of the idoneal type discussed in A139827. The remaining 21 begin at A140613 and end here. The cross-references section lists the quadratic forms in the same order as tables 1-6 in Voight's paper. Note that A107169 and A139831 are in the same equivalence class.
In base 12, the sequence is 7, 87, X7, 167, 267, 327, 347, 427, 507, 587, 687, 747, 767, 907, 927, 9X7, X07, X87, XX7, E67, 1047, 12X7, 13X7, 1467, 1547, 1647, 1727, 1807, 1E27, 2007, 20X7, 2107, 21X7, 2267, 2287, 2347, 2367, 2607, 2687, 26X7, 2787, 2847, where X is for 10 and E is for 11. Moreover, the discriminant is X00 and that all elements are {7, 87, X7, 167, 187, 247} mod 260. - Walter Kehowski, May 31 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
John Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

Cf. A033205, A007519, A068228, A107135, A107144, A107152, A107151.
Cf. A107181, A139502, A139856, A139854, A139874, A139877, A139897.
Cf. A140613, A140614. A140615, A139923, A140616-A140619, A139988.
Cf. A139993, A140008, A140010, A140620-A140632, A033212, A102273.
Cf. A107007, A107003, A139830, A107169, A139831, A141373, A139847.
Cf. A139850, A139855, A139860, A139879, A139880, A139924, A139990.
Cf. A139998, A139992, A139996, A140003, A140013, A107006, A139858.
Cf. A107008, A140633, A139991, A139857, A139859, A139878, A007645, A002313.

Union[QuadPrimes2[7, 4, 52, 10000], QuadPrimes2[7, -4, 52, 10000]] (* see A106856 *)

A168539 Terms of A123239 which are prime in Z(i), Z(rho) and Z(sqrt(2)).

Original entry on oeis.org

11, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 827, 947, 971, 1019, 1091, 1163, 1187, 1259, 1283, 1307, 1427, 1451, 1499, 1523, 1571, 1619, 1667, 1787, 1811, 1907, 1931, 1979, 2003, 2027, 2099, 2243, 2267, 2339

Offset: 1

A.K. Devaraj, Nov 29 2009

nonn

Cf. A002145, A003627, A003629, A123239, A107007, A168454.

MangammalQ[p_] := Block[{k = 3}, While[k > 2, k = Mod[3 k, p]]; k != 2];
A168539 = Select[Prime[Range[350]], MangammalQ[#] && Mod[#, 24] == 11 &] (* Ray Chandler, Jul 21 2011 *)

Corrected and extended by Ray Chandler, Jul 21 2011

A018188 The $620 prime list.

Original entry on oeis.org

5003, 5987, 6563, 9803, 10427, 11027, 11867, 16763, 19403, 22283, 22907, 24923, 25667, 29867, 35747, 40427, 40763, 41243, 42083, 49307, 54323, 54347, 57203, 57347, 66587, 67307, 73883, 78203, 84347, 104003, 112067, 121403

Offset: 1

N. J. A. Sloane, Jon Grantham (grantham(AT)math.uga.edu)

Numbers are 11 mod 24.
Jon Grantham: "I strongly believe that some sub-product of these primes is a Carmichael number and a Lucas pseudoprime for the Fibonacci sequence, and also is 2 or 3 mod 5."
In particular the above conjecture implies that there is a BPSW pseudoprime smaller than 10^25286. - Charles R Greathouse IV, Sep 28 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..2030
Jon Grantham, Pseudoprimes/Probable Primes

Subsequence of A107007 (and hence A000040).

b-file from Charles R Greathouse IV, Aug 28 2010

A214151 Numbers k from the set == 5 (mod 6) with the property that 3^((3*k-1)/2) == 3 (mod k) and 2^((k-1)/2) == (k-1) (mod k).

Original entry on oeis.org

11, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 827, 947, 971, 1019, 1091, 1163, 1187, 1259, 1283, 1307, 1427, 1451, 1499, 1523, 1571, 1619, 1667, 1787, 1811, 1907, 1931, 1979, 2003, 2027, 2099, 2243, 2267

Offset: 1

Alzhekeyev Ascar M, Jul 05 2012

nonn

All composites in this sequence are 2-pseudoprimes, see A001567, and strong pseudoprimes to base 2, A001262.
The subsequence of these composites begins: 1441091, 3587553971, 4528686251, 23260036451, 47535120323, 61070250323, 90474845819, 143193768587, 162016315907, 173868807611, 180998962187, 238364070323, 285370693931, 298577370323, ...
Perhaps this sequence contains all the terms of the sequence A107007 or A168539.

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

Subsequence of A176997.

Cf. A003629, A006970, A175865.

isA214151 := proc(n)
    if (n mod 6 = 5) and modp(2 &^ ((n-1)/2),n)  = n-1 and modp(3 &^ ((3*n-1)/2),n)  = 3 then
        true;
    else
        false;
    end if;
end proc:
for n from 5 by 6 do
    if isA214151(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Jul 20 2012

Select[Range[5,2500,6],PowerMod[3,(3#-1)/2,#]==3&&PowerMod[2,(#-1)/2,#] == #-1&] (* Harvey P. Dale, Mar 14 2022 *)

for(n=0, 200, b=6*n+5; if(Mod(3, b)^((3*b-1)/2)==3, if(Mod(2, b)^((b-1)/2)==b-1 , print1(b, ", "))));

A139527 Numbers n such that numbers 24n+5 are primes.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 11, 12, 13, 16, 19, 21, 23, 27, 28, 29, 32, 33, 34, 39, 42, 44, 46, 49, 51, 53, 54, 57, 62, 67, 68, 71, 72, 78, 79, 81, 82, 83, 86, 89, 92, 93, 96, 97, 98, 99, 103, 106, 109, 112, 114, 116, 118, 119, 121, 123, 134, 141, 142, 144, 147, 148, 149, 153, 154

Offset: 1

Artur Jasinski, Apr 25 2008

nonn

Numbers n such that:
24n+1 is prime see A111174, primes 24n+1 see A107008
24n+5 is prime see A139527, primes 24n+5 see A107003
24n+7 is prime see A139483, primes 24n+7 see A107006
24n+11 is prime A139528, primes 24n+11 see A107007
24n+13 is prime see A139529, primes 24n+13 see A139530
24n+17 is prime see A139531, primes 24n+17 see A107181
24n+19 is prime see A139532, primes 24n+19 see A141373
24n+23 is prime see A131210, primes 24n+23 see A134517

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

a = {}; Do[If[PrimeQ[24 n + 5], AppendTo[a, n]], {n, 0, 200}]; a
Select[Table[(Prime[n]-5)/24,{n,800}],IntegerQ] (* Harvey P. Dale, Feb 25 2016 *)

A273618 Numbers m = 2*k+1 where k is odd with the property that 3^k mod m = 1 and k^k mod m = 1.

Original entry on oeis.org

11, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 827, 947, 971, 1019, 1091, 1163, 1187, 1259, 1283, 1307, 1427, 1451, 1499, 1523, 1571, 1619, 1667, 1787, 1811, 1907, 1931, 1979, 2003, 2027, 2099, 2243, 2267

Offset: 1

Alzhekeyev Ascar M, May 26 2016

nonn

All composites in this sequence are 2-pseudoprimes, see A001567, and strong pseudoprimes to base 2, A001262.
The subsequence of these composites begins: 143193768587, 440097066011, 1188059560451, 1392770336147, 1640446291859, 2526966350771, 3639120872171, 3989703695867, 4202422108523, ....
Perhaps this sequence contains all the terms of the sequence A107007 (except 3) or A168539.

			m=131; 131=2*65+1; 3^65 mod 131 = 1 and 65^65 mod 131 = 1.

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

Subsequence of A176997.

Cf. A001262, A001567, A107007, A168539, A214151.

filter:= proc(n) local k;
  k:= (n-1)/2;
  3 &^ k mod n = 1 and k &^ k mod n = 1
end proc:
select(filter, [seq(i,i=3..3000, 4)]); # Robert Israel, Nov 28 2019

2#+1&/@Select[Range[1,1200,2],PowerMod[3,#,2#+1]==PowerMod[ #,#,2#+1] == 1&] (* Harvey P. Dale, May 05 2022 *)

cp's OEIS Frontend

A140633 Primes of the form 7x^2+4xy+52y^2.

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A168539 Terms of A123239 which are prime in Z(i), Z(rho) and Z(sqrt(2)).

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A018188 The $620 prime list.

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A214151 Numbers k from the set == 5 (mod 6) with the property that 3^((3*k-1)/2) == 3 (mod k) and 2^((k-1)/2) == (k-1) (mod k).

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A139527 Numbers n such that numbers 24n+5 are primes.

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A273618 Numbers m = 2*k+1 where k is odd with the property that 3^k mod m = 1 and k^k mod m = 1.

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