A007519 -id:A007519 - cp's OEIS frontend

A033200 Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.

3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(-2)). - N. J. A. Sloane, Dec 25 2017
Fermat knew of the relationship between a prime being congruent to 1 or 3 mod 8 and its being the sum of a square and twice a square, and claimed to have a firm proof of this fact. These numbers are not primes in Z[sqrt(-2)], as they have x - y sqrt(-2) as a divisor. - Alonso del Arte, Dec 07 2012
Terms m in A047471 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012
This sequence gives the primes p which satisfy norm(rho(p)) = + 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For the minimal polynomial C see A187360. For p == 1 (mod 8) the norm is C(p, 0) (see a comment on 4*A005123) and for p == 3 (mod 8) the norm is -C(p, 0) (see a comment on A186297). For the primes with norm(rho(p)) = -1 see A003628. - Wolfdieter Lang, Oct 24 2013
If p is a member then it has a unique representation as x^2+2y^2 [Frei, Theorem 3]. - N. J. A. Sloane, May 30 2014
Primes that are the quarter perimeter of a Heronian triangle. Such primes are unique to the Heronian triangle (see Yiu link). - Frank M Jackson, Nov 30 2014

			Since 11 is prime and 11 == 3 (mod 8), 11 is in the sequence. (Also 11 = 3^2 + 2 * 1^2 = (3 + sqrt(-2))(3 - sqrt(-2)).)
Since 17 is prime and 17 == 1 (mod 8), 17 is in the sequence.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from T. D. Noe]
G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), p. 56.
Zak Seidov, Table of n, a(n), x and y for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Paul Yiu, CRUX, Problem 2331, Proposed by Paul Yiu
Paul Yiu and Jill S. Taylor, CRUX, Problem 2331, Solution pp 185-186
Index to sequences related to decomposition of primes in quadratic fields

Cf. A033203.

a033200 n = a033200_list !! (n-1)
a033200_list = filter ((== 1) . a010051) a047471_list
-- Reinhard Zumkeller, Dec 29 2012

[p: p in PrimesUpTo(600) | p mod 8 in [1, 3]]; // Vincenzo Librandi, Aug 04 2012

Rest[QuadPrimes2[1, 0, 2, 10000]] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,3},Mod[#,8]]&] (* Harvey P. Dale, Jun 09 2017 *)

is(n)=n%8<4 && n%2 && isprime(n) \\ Charles R Greathouse IV, Feb 09 2017

a(n) = A033203(n+1). - Zak Seidov, May 29 2014
A007519 UNION A007520. - R. J. Mathar, Jun 09 2020
L(-2, a(n)) = +1, n >= 1, with the Legendre symbol L. -Wolfdieter Lang, Jul 24 2024

A133870 Primes of the form 32*n + 1.

Original entry on oeis.org

97, 193, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1249, 1409, 1601, 1697, 1889, 2017, 2081, 2113, 2273, 2593, 2657, 2689, 2753, 3041, 3137, 3169, 3329, 3361, 3457, 3617, 4001, 4129, 4289, 4481, 4513, 4673, 4801, 4993, 5153, 5281, 5441, 5569

Offset: 1

Zak Seidov, Sep 27 2007

Corresponding n's: 3, 6, 8, 11, 14, 18, 20, 21, 24, 29, 36, 38, 39, ... (A133869).
These primes p are the only ones with the property that for every integer m from interval [0,p) with the Hamming  distance D(m,p) = 2 or 3, there exists an integer h from (m,p) with D(m,h) = D(m,p). - Vladimir Shevelev, Apr 19 2012
Primes p such that p XOR 30 = p + 30. - Brad Clardy, Jul 22 2012
Odd primes p such that -1 is a 16th power mod p. - Eric M. Schmidt, Mar 27 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A133869; A065091, A002144, A007519, A094407, A142925, A208177, A208178, A076339.

a133870 n = a133870_list !! (n-1)
a133870_list = filter ((== 1) . a010051) [1,33..]
-- Reinhard Zumkeller, Mar 06 2012

[p: p in PrimesUpTo(12000) | p mod 32 eq 1 ]; // Vincenzo Librandi, Aug 18 2012

Select[32*Range[175] + 1, PrimeQ] (* Alonso del Arte, Jul 24 2012 *)
Select[Prime[Range[4000]],MemberQ[{1},Mod[#,32]]&] (* Vincenzo Librandi, Aug 18 2012 *)

A334826 Decimal expansion of Product_{primes p==1 mod 8} (1 - 4/p)*((p + 1)/(p - 1))^2.

Original entry on oeis.org

9, 5, 6, 9, 4, 5, 3, 4, 7, 8, 5, 1, 6, 0, 1, 1, 8, 3, 4, 3, 6, 9, 6, 7, 0, 5, 7, 2, 7, 3, 8, 9, 1, 8, 2, 8, 7, 5, 3, 1, 7, 4, 9, 7, 7, 2, 9, 1, 3, 9, 1, 4, 7, 8, 9, 0, 5, 4, 3, 2, 6, 0, 4, 2, 4, 6, 0, 1, 7, 0, 1, 6, 4, 4, 4, 8, 8, 8, 8, 5, 9, 4, 8, 1, 4, 4, 0, 5, 1, 2, 0, 3, 9, 0, 7, 9, 5

Offset: 0

Vaclav Kotesovec, May 13 2020

Note that Product_{k>=1} (4*k + 1)^2 * (8*k - 3) / (16 * k^2 * (8*k + 1)) = 2^(1/4) * Gamma(1/8)^2 / (sqrt(Pi) * Gamma(1/4)^3) = 0.79906817873784592665354...

			0.9569453478516011834369670572738918287531749772913914789...

Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019.
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A007519, A210630.

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Zs[8, 1, 4]/Z[8, 1, 2]^2, digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)
(* -------------------------------------------------------------------------- *)
(* second program, more general *)
$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (1 - 4/p)*((p + 1)/(p - 1))^2;
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*(P[8, 1, m] - 1/17^m); sump = sump + difp; m++];
RealDigits[Chop[N[f[17]*Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

A120305 a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * (i+j)!/(i!j!).

Original entry on oeis.org

1, 1, 3, 9, 31, 111, 407, 1513, 5679, 21471, 81643, 311895, 1196131, 4602235, 17757183, 68680169, 266200111, 1033703055, 4020716123, 15662273839, 61092127491, 238582873475, 932758045123, 3650336341239, 14298633670931

Offset: 0

Alexander Adamchuk, Jul 14 2006

nonn

p divides a((p+1)/2) for prime p = 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, ... (A033200: primes congruent to {1, 3} mod 8; or, odd primes of the form x^2 + 2*y^2).
p divides a((p-3)/2) for prime p = 17, 41, 73, 89, 97, 113, 137, ... (A007519: primes of the form 8n+1).
Essentially the same as partial sums of A072547. - Seiichi Manyama, Jan 30 2023

Seiichi Manyama, Table of n, a(n) for n = 0..1664 (terms 0..200 from Vincenzo Librandi)

Cf. A000108, A006134, A033200, A007519, A092785, A307354.

Cf. A072547, A371798.

Table[Sum[Sum[(-1)^(i+j)*(i+j)!/(i!j!),{i,0,n}],{j,0,n}],{n,0,50}]

a(n) = sum(i=0, n, sum(j=0, n, (-1)^(i+j) * (i+j)!/(i!*j!))); \\ Michel Marcus, Apr 02 2019

a(n) = sum(i=0, 2*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^2, i)); \\ Seiichi Manyama, May 20 2019

my(N=30, x='x+O('x^N)); Vec((1+sqrt(1-4*x))/(sqrt(1-4*x)*(1-x)*(3-sqrt(1-4*x)))) \\ Seiichi Manyama, Jan 30 2023

a(n) = Sum_{j=0..n} Sum_{i=0..n} (-1)^(i+j)*(i+j)!/(i!j!).
Recurrence: 2*n*(3*n-5)*a(n) = 3*(9*n^2 - 19*n + 8)*a(n-1) - 3*(n-1)*(3*n-4)*a(n-2) - 2*(2*n-3)*(3*n-2)*a(n-3). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 4^(n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 13 2013
G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( (1 - x *c(x))/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023
From Seiichi Manyama, Apr 06 2024: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k-1,n-3*k).
a(n) = [x^n] 1/((1+x^3) * (1-x)^n). (End)

A139487 Numbers k such that 8k + 7 is prime.

Original entry on oeis.org

0, 2, 3, 5, 8, 9, 12, 15, 18, 20, 23, 24, 27, 29, 32, 33, 38, 44, 45, 47, 53, 54, 57, 59, 60, 62, 74, 75, 78, 80, 89, 90, 92, 93, 102, 104, 107, 110, 113, 114, 120, 122, 123, 128, 129, 132, 135, 137, 143, 152, 153, 159, 162, 164, 165, 170, 174, 177, 179, 180, 183, 185

Offset: 1

Artur Jasinski, Apr 23 2008

For numbers k such that:
8k+1 is prime see A005123, primes see A007519;
8k+3 is prime see A005124, primes see A007520;
8k+5 is prime see A105133, primes see A007521;
8k+7 is prime see A139487, primes see A007522.
8k + 7 divides A000225(4k+3). - Jinyuan Wang, Mar 08 2019

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

Cf. A000225, A007519, A007520, A007521, A007522, A005123, A005124, A105133.

[n: n in [0..200] | IsPrime(8*n+7)]; // Vincenzo Librandi, Jun 25 2014

a = {}; Do[If[PrimeQ[8 n + 7], AppendTo[a, n]], {n, 0, 300}]; a
Select[Range[0,200],PrimeQ[8#+7]&] (* Harvey P. Dale, Oct 10 2012 *)

is(n)=isprime(8*n+7) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A007522(n) - 7)/8, n >= 1.

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

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

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A076339 Primes of the form 512*k+1.

Original entry on oeis.org

7681, 10753, 11777, 12289, 13313, 15361, 17921, 18433, 19457, 23041, 25601, 26113, 32257, 36353, 37889, 39937, 40961, 45569, 50177, 51713, 58369, 59393, 61441, 64513, 65537, 67073, 70657, 76289, 76801, 79873, 80897, 81409, 83969

Offset: 1

Reinhard Zumkeller, Oct 07 2002

Odd primes p such that -1 is a 256th power mod p. - Eric M. Schmidt, Mar 27 2014

			A076338(15) = 512*15+1 = a(1) = 7681 = A000040(974);
A076338(21) = 512*21+1 = a(2) = 10753 = A000040(1311);
a(38) - a(37) = 95233 - 87553 = 7680 = a(1)-1.

M. Kraitchik, Theorie des Nombres, Gauthier-Villars (I. 1922, II. 1929).
M. Kraitchik, Recherches sur la theorie des nombres, Gauthier-Villars (1924).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A076338, A065091, A002144, A007519, A094407, A133870, A142925, A208177, A208178.

a076339 n = a076339_list !! (n-1)
a076339_list = filter ((== 1) . a010051) [1,513..]
-- Reinhard Zumkeller, Mar 06 2012

Select[512 Range[164] + 1, PrimeQ] (* Bruno Berselli, Feb 23 2012 *)

forprimestep(p=7681,83969,512, print1(p", ")) \\ Charles R Greathouse IV, Nov 01 2022

a(n) ~ 256n log n. - Charles R Greathouse IV, Nov 01 2022

A142925 Primes congruent to 1 mod 64.

Original entry on oeis.org

193, 257, 449, 577, 641, 769, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4289, 4481, 4673, 4801, 4993, 5441, 5569, 5953, 6337, 6529, 6977, 7297, 7489, 7681, 7873, 7937, 8513, 8641, 9281, 9473, 9601, 9857, 10177, 10369, 10433, 10753, 11329

Offset: 1

N. J. A. Sloane, Jul 11 2008

Odd primes p such that -1 is a 32nd power mod p. - Eric M. Schmidt, Mar 27 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A065091, A002144, A007519, A094407, A133870, A208177, A208178, A076339.

a142925 n = a142925_list !! (n-1)
a142925_list = filter ((== 1) . a010051) [1,65..]
-- Reinhard Zumkeller, Mar 06 2012

[p: p in PrimesUpTo(12000) | p mod 64 eq 1 ] ; // Vincenzo Librandi, Sep 06 2012

lst={};Do[p=Prime[n];If[Mod[p,64]==1,AppendTo[lst,p]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
Select[Prime[Range[1800]], MemberQ[{1}, Mod[#, 64]] &] (* Vincenzo Librandi, Sep 06 2012 *)
Select[Range[1,12000,64],PrimeQ] (* Harvey P. Dale, Dec 25 2023 *)

is(n)=isprime(n) && n%64==1 \\ Charles R Greathouse IV, Jul 01 2016

A208177 Primes of the form 128*k + 1.

Original entry on oeis.org

257, 641, 769, 1153, 1409, 2689, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857, 10369, 10753, 11393, 11777, 12161, 12289, 13313, 13441, 13697, 14081, 14593, 15233, 15361, 16001, 17921, 18049, 18433, 19073, 19457, 19841, 20353, 21121, 21377

Offset: 1

Bruno Berselli, Feb 25 2012

Odd primes p such that -1 is a 64th power mod p. - Eric M. Schmidt, Mar 27 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A065091, A002144, A007519, A094407, A133870, A142925, A208178, A076339.

a208177 n = a208177_list !! (n-1)
a208177_list = filter ((== 1) . a010051) [1,129..]
-- Reinhard Zumkeller, Mar 06 2012

[ p: p in PrimesUpTo(22000) | p mod 128 eq 1 ];

Mathematica
```
Select[128 Range[167] + 1, PrimeQ]
```

for(n=1,167,r=128*n+1; if(isprime(r), print1(r", ")));

a(n) ~ 64n log n. - Charles R Greathouse IV, Nov 01 2022

A014754 Primes p == 1 mod 8 such that 2 and -2 are both 4th powers (one implies other) mod p.

Original entry on oeis.org

73, 89, 113, 233, 257, 281, 337, 353, 577, 593, 601, 617, 881, 937, 1033, 1049, 1097, 1153, 1193, 1201, 1217, 1249, 1289, 1433, 1481, 1553, 1601, 1609, 1721, 1753, 1777, 1801, 1889, 1913, 2089, 2113, 2129, 2273, 2281, 2393, 2441, 2473, 2593, 2657, 2689

Offset: 1

Warren D. Smith

nonn

Primes p such that x^4 == 2 has more than two (in fact four) solutions mod p. This is the sequence of terms common to A040098 (primes p such that x^4 == 2 has a solution mod p) and A007519 (primes of form 8n+1). Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 == 2 iff p - i is a solution mod p of x^4 == 2, thus the sum of first and fourth solution is p and so is the sum of second and third solution. The solutions are given in A065909, A065910, A065911 and A065912. - Klaus Brockhaus, Nov 28 2001

Primes of the form x^2+64y^2. - T. D. Noe, May 13 2005

N. J. A. Sloane and Vincenzo Librandi, Table of n, a(n) for n = 1..9769 (the first 1000 terms were found by Vincenzo Librandi)

Cf. A040098, A007519, A014754, A007522, A065909, A065910, A065911, A065912, A070179.

A014754(m) = local(p,s,x,z); forprime(p = 3,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); z = matsize(s)[2]; if(z>2,print1(p,",")))

{a(n) = local(m, c, x); if( n<1, 0, c = 0; m = 1; while( cMichael Somos, Mar 22 2008 */

forprime(p=1, 9999, p%8==1&&ispower(Mod(2, p), 4)&&print1(p", ")) \\ M. F. Hasler, Feb 18 2014

is_A014754(p)={p%8==1&&ispower(Mod(2, p), 4)&&isprime(p)} \\ M. F. Hasler, Feb 18 2014

Removed erroneous Mma program; extended b-file using first PARI program of M. F. Hasler. - N. J. A. Sloane, Jun 06 2014

cp's OEIS Frontend

A033200 Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

A133870 Primes of the form 32*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

A334826 Decimal expansion of Product_{primes p==1 mod 8} (1 - 4/p)*((p + 1)/(p - 1))^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A120305 a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * (i+j)!/(i!j!).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A139487 Numbers k such that 8k + 7 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A141849 Primes congruent to 1 mod 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Formula

A076339 Primes of the form 512*k+1.

Views

Author