A033200 -id:A033200 - cp's OEIS frontend

A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

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, 521, 523, 547, 563, 569, 571, 577, 587, 593, 601, 617, 619, 641, 643, 659, 673, 683

Offset: 1

N. J. A. Sloane

Sequence naturally partitions into two sequences: all primes p with ord_p(-2) odd (A163183, the primes dividing 2^j +1 for some odd j) and certain primes p with ord_p(-2) even (A163185). - Christopher J. Smyth, Jul 23 2009

Terms m in A047476 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012

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

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002332, A002333.
Cf. A039706, A003628 (complement with respect to A000040).
Primes in A002479.
Cf. A051100 (see Mathar's comment).
Apart from leading term the same as A033200.

a033203 n = a033203_list !! (n-1)
a033203_list = filter ((== 1) . a010051) a047476_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012

[p: p in PrimesUpTo(600) | p mod 8 in [1..3]]; // Vincenzo Librandi, Aug 11 2012

[p: p in PrimesUpTo(800) | NormEquation(2,p) eq true]; // Bruno Berselli, Jul 03 2016

QuadPrimes2[1, 0, 2, 10000] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,2,3},Mod[#,8]]&] (* Harvey P. Dale, Mar 16 2013 *)

is(n)=isprime(n) && issquare(Mod(-2,n)) \\ Charles R Greathouse IV, Nov 29 2016

a(n) = A002332(n) + 2*A002333(n)^2. - Zak Seidov, May 29 2014

A048861 a(n) = n^n - 1.

Original entry on oeis.org

0, 3, 26, 255, 3124, 46655, 823542, 16777215, 387420488, 9999999999, 285311670610, 8916100448255, 302875106592252, 11112006825558015, 437893890380859374, 18446744073709551615, 827240261886336764176, 39346408075296537575423, 1978419655660313589123978

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com)

From Alexander Adamchuk, Jan 22 2007: (Start)
a(n) is divisible by (n-1).
Corresponding quotients are a(n)/(n-1) = {1,3,13,85,781,9331, ...} = A023037(n).
p divides a(p-1) for prime p.
p divides a((p-1)/2) for prime p = {3,11,17,19,41,43,59,67,73,83,89,97,...} = A033200 Primes congruent to {1, 3} mod 8; or, odd primes of form x^2+2*y^2.
p divides a((p-1)/3) for prime p = {61,67,73,103,151,193,271,307,367,...} = A014753 3 and -3 are both cubes (one implies other) mod these primes p=1 mod 6.
p divides a((p-1)/4) for prime p = {5,13,17,29,37,41,53,61,73,...} = A002144 Pythagorean primes: primes of form 4n+1.
p divides a((p-1)/5) for prime p = {31,191,251,271,601,641,761,1091,...}.
p divides a((p-1)/6) for prime p = {7,241,313,337,409,439,607,631,727,751,919,937,...}. (End)
For n > 1, a(n) is largest number that can be represented using n digits in the base-n number system. - Chinmaya Dash, Mar 31 2022

			For n=3, a(n) = 3^3 - 1 = 27 - 1 = 26. - _Michael B. Porter_, Nov 12 2017

M. Le, Primes in the sequences n^n+1 and n^n-1, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 156-157.

G. C. Greubel, Table of n, a(n) for n = 1..385
F. Smarandache, Only Problems, Not Solutions!

Cf. A000312, A048860, A023037, A033200, A014753, A002144.

[ n^n-1: n in [1..25]]; // Vincenzo Librandi, Dec 29 2010

Table[n^n - 1, {n, 1, 50}] (* G. C. Greubel, Nov 10 2017 *)

a(n)=n^n-1 \\ Charles R Greathouse IV, Feb 24 2012

E.g.f.: 1/(1+LambertW(-x)) - exp(x). - Vaclav Kotesovec, Dec 20 2014

Extended (and corrected) by Patrick De Geest, Jul 15 1999

A003628 Primes congruent to {5, 7} mod 8.

Original entry on oeis.org

5, 7, 13, 23, 29, 31, 37, 47, 53, 61, 71, 79, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 199, 223, 229, 239, 263, 269, 271, 277, 293, 311, 317, 349, 359, 367, 373, 383, 389, 397, 421, 431, 439

Offset: 1

N. J. A. Sloane, Mira Bernstein

Inert rational odd primes in the field Q(sqrt(-2)).
Primes p such that p XOR 5 = p - 5. - Brad Clardy, Jul 22 2012
Terms m in A047566 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 p == 5 (mod 8) the norm is C(p, 0) (see a comment on 2*A230076) and for p == 7 (mod 8) the norm is -C(p, 0) (see a comment on A186302). For the primes with norm(rho(p)) = +1 see A033200. - Wolfdieter Lang, Oct 24 2013

H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to sequences related to decomposition of primes in quadratic fields

Cf. A000040, A039706, A033203 (complement with respect to A000040).

a003628 n = a003628_list !! (n-1)
a003628_list = filter ((== 1) . a010051) a047566_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012

[ p: p in PrimesUpTo(600) | p mod 8 in {5, 7}]; // Vincenzo Librandi, Aug 22 2012

Select[Prime[Range[200]],MemberQ[{5,7},Mod[#,8]]&] (* Harvey P. Dale, Oct 24 2011 *)

{a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( -2, m )==-1, cnt++ )); m} /* Michael Somos, Aug 14 2012 */

a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 24 2023

A163183 Primes dividing 2^j + 1 for some odd j.

Original entry on oeis.org

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 281, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 617, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1033, 1049, 1051, 1091, 1097

Offset: 1

Christopher J. Smyth, Jul 22 2009

Also the primes p for which ord_p(-2) is odd, as (-2)^j == 1 (mod p).
All such p are = 1 or 3 mod 8, so sequence is subsequence of A033200, as (-2)^{j+1} == -2 (mod p) implies that (-2/p) = 1, p == 1 or 3 (mod 8).
Claim: Sequence contains all primes = 3 mod 8, so contains A007520 as a subsequence.
Proof: If p = 8r + 3 then 2^{4r+1} == 1 or -1 (mod p). If former, then (2^{2r+1})^2 == 2 (mod p), (2/p) = 1, only true for p == 1 or 7 (mod 8). So p | 2^{4r+1} + 1.
Also contains some primes == 1 (mod 8), given in A163184. So sequence is a union of A007520 and A163184.
Claim: For every p in sequence and every 2^k, the equation x^{2^k} == -2 (mod p) is soluble. Hence sequence is a subsequence of A033203 (k=1), A051071 (k=2), A051073 (k=3), A051077 (k=4), A051085 (k=5), A051101 (k=6), ....
Proof: Put x == (-2)^u (mod p). Then using (-2)^j == 1 (mod p), we can solve x^{2^k} == -2 (mod p) if can find u and v such that u*2^k + v*j = 1, possible as gcd(2^k, j) = 1.
From Jianing Song, Jun 22 2025: (Start)
The multiplicative order of -2 modulo a(n) is A385228(n).
Contained in primes congruent to 1 or 3 modulo 8 (primes p such that -2 is a quadratic residue modulo p, A033200), and contains primes congruent to 3 modulo 8 (A007520).
Conjecture: this sequence has density 7/24 among the primes (see A014663). (End)

			11 is in sequence as 11 | 2^5 + 1; 281 (smallest element of the sequence == 1 (mod 8)) is in the sequence as 281 | 2^35 + 1.

Jianing Song, Table of n, a(n) for n = 1..10000

Sequence is a union of A007520 and A163184.
Subsequence of A033200. Contains A007520 as a subsequence.
Cf. A385228 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), this sequence (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

with(numtheory):A:=3:p:=3: for c to 500 do p:=nextprime(p);if order(-2,p) mod 2=1 then A:=A,p;;fi;od:A;

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-2, #]] &] (* Paolo Xausa, Jun 30 2025 *)

lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2, print1(p, ", "))); \\ Jinyuan Wang, Mar 23 2020

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)

A047471 Numbers that are congruent to {1, 3} mod 8.

Original entry on oeis.org

1, 3, 9, 11, 17, 19, 25, 27, 33, 35, 41, 43, 49, 51, 57, 59, 65, 67, 73, 75, 81, 83, 89, 91, 97, 99, 105, 107, 113, 115, 121, 123, 129, 131, 137, 139, 145, 147, 153, 155, 161, 163, 169, 171, 177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233

Offset: 1

N. J. A. Sloane

			For n=2, a(2) = 8*2-1-12 = 3;
For n=3, a(3) = 8*3-3-12 = 9;
For n=4, a(4) = 8*4-9-12 = 11. - _Vincenzo Librandi_, Aug 06 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A017077 and A017101.

Cf. A033200 (primes).

a047471 n = a047471_list !! (n-1)
a047471_list = [n | n <- [1..], mod n 8 `elem` [1,3]]
-- Reinhard Zumkeller, Dec 29 2012

[4*(n-1)-(-1)^n : n in [1..80]]; // Wesley Ivan Hurt, Apr 28 2017

A047471:=n->4*n - 4 - (-1)^n; seq(A047471(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2014

Table[4 n - 4 - (-1)^n, {n, 100}] (* Wesley Ivan Hurt, Jan 30 2014 *)
#+{1,3}&/@(8*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,3,9},60] (* Harvey P. Dale, Jan 05 2017 *)

G.f.: x*(1+2*x+5*x^2)/((1+x)*(1-x)^2). - Paul Barry, Apr 10 2008
a(n) = 4*(n-1)-(-1)^n. - Rolf Pleisch, Aug 04 2009
a(n) = 8*n-a(n-1)-12, with a(1)=1. - Vincenzo Librandi, Aug 06 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + sqrt(2)*log(sqrt(2)+1)/4. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

A341784 Norms of prime elements in Z[sqrt(-2)], the ring of integers of Q(sqrt(-2)).

Original entry on oeis.org

2, 3, 11, 17, 19, 25, 41, 43, 49, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 169, 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, 521, 523, 529, 547, 563

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[sqrt(-2)], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes congruent to 1, 2, 3 modulo 8 and the squares of primes congruent to 5, 7 modulo 8.
For primes p == 1, 3 (mod 8), there are two distinct ideals with norm p in Z[sqrt(2)], namely (x + y*sqrt(-2)) and (x - y*sqrt(-2)), where (x,y) is a solution to x^2 + 2*y^2 = p; for p = 2, (sqrt(-2)) is the unique ideal with norm p; for p == 5, 7 (mod 8), (p) is the only ideal with norm p^2.

			norm(1 + sqrt(-2)) = norm(1 + sqrt(-2)) = 3;
norm(3 + sqrt(-2)) = norm(3 + sqrt(-2)) = 11;
norm(3 + 2*sqrt(-2)) = norm(3 + 2*sqrt(-2)) = 17;
norm(1 + 3*sqrt(-2)) = norm(1 + 3*sqrt(-2)) = 19.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A188510, A033203, A033200, A003628.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A002325.
The total number of elements with norm n is given by A033715.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), this sequence (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341784(n) = my(disc=-8); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A363409 a(n) = the real part of Product_{k = 1..n} (1 + k*sqrt(-2)).

Original entry on oeis.org

1, 1, -3, -21, 27, 927, 387, -78111, -211167, 10887129, 61228629, -2278564101, -20995423317, 669639978711, 9055735268283, -263207953694367, -4900375484030367, 133357760824723281, 3278778524907635277, -84617763517115570709, -2669012118280627019109

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes. Numerical calculation suggests that a similar division also holds in this case.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {2, 5, 13, 23, 29, 31, 47, 53, 61, 71, 101, ...}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 or 3 (mod 8), i.e., rational primes that split in the field extension Q(sqrt(-2)) of Q. See A033200.
Moll's conjecture 5.5 extends to this sequence and takes the form: for prime p == 1 or 3 (mod 8), the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes begins {7, 37, 79, 103, ...}.
Taken together, the type 1 and type 3 primes appear to consist of all primes p == 5 or 7 (mod 8), that is, the rational primes that remain inert in the field extension Q(sqrt(-2)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-2)). See A003628.

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 0..100] = [0, 0, 1, 1, 3, 2, 2, 3, 5, 5, 4, 5, 5, 6, 6, 6, 9, 9, 9, 10, 10, 10, 11, 11, 11, 13, 13, 13, 14, 14, 14, 15, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 19, 22, 22, 22, 23, 23, 23, 24, 24, 24, 26, 26, 26, 27, 27, 27, 28, 28, 28, 30, 30, 30, 31, 31, 31, 32, 32, 32, 37, 36, 36, 40, 37, 37, 38, 38, 38, 40, 40, 40, 41, 41, 41, 42, 42, 42, 47, 44, 44, 46, 45, 45, 46, 46, 46, 49, 49, 49, 50].
Note that v_3(a(100)) = 50 = 100/(3 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 0..101] = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1], showing the oscillatory behavior for type 3 primes conjectured above. It appears that v_7(a(7*n+3)) = 1 otherwise v_7(a(n)) = 0.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A363410 - A363416.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else ( (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Table[Re[Product[1+k*Sqrt[-2], {k, 0, n}]], {n, 0, 20}] (* James C. McMahon, Jan 28 2024 *)

a(n) = Sum_{k = 0..floor((n+1)/2)} (-2)^k*Stirling1(n+1, n+1-2*k).
a(n+1)/a(n) = 1 - (2*n + 2)*1/sqrt(2)*tan( Sum_{k = 1..n} arctan(sqrt(2)*k) ).
(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) with a(0) = 1 and a(1) = 1.

A296937 Rational primes that decompose in the field Q(sqrt(13)).

Original entry on oeis.org

3, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641

Offset: 1

N. J. A. Sloane, Dec 26 2017

Is this the same sequence as A141188 or A038883? - R. J. Mathar, Jan 02 2018
From Jianing Song, Apr 21 2022: (Start)
Primes p such that Kronecker(13, p) = Kronecker(p, 13) = 1, where Kronecker() is the Kronecker symbol. That is to say, primes p that are quadratic residues modulo 13.
Primes p such that p^6 == 1 (mod 13).
Primes p == 1, 3, 4, 9, 10, 12 (mod 13). (End)

Cf. A011583 (kronecker symbol modulo 13), A038883.
Rational primes that decompose in the quadratic field with discriminant D: A139513 (D=-20), A191019 (D=-19), A191018 (D=-15), A296920 (D=-11), A033200 (D=-8), A045386 (D=-7), A002144 (D=-4), A002476 (D=-3), A045468 (D=5), A001132 (D=8), A097933 (D=12), this sequence (D=13), A296938 (D=17).
Cf. A038884 (inert rational primes in the field Q(sqrt(13))).

Load the Maple program HH given in A296920. Then run HH(13, 200); This produces A296937, A038884, A038883.

isA296937(p) = isprime(p) && kronecker(p,13) == 1 \\ Jianing Song, Apr 21 2022

Equals A038883 \ {13}. - Jianing Song, Apr 21 2022

A227897 Numbers k such that k^2 + 2 is not squarefree.

Original entry on oeis.org

4, 5, 13, 14, 19, 22, 23, 24, 31, 32, 40, 41, 49, 50, 58, 59, 63, 67, 68, 71, 76, 77, 85, 86, 94, 95, 102, 103, 104, 112, 113, 121, 122, 130, 131, 139, 140, 148, 149, 157, 158, 166, 167, 175, 176, 184, 185, 193, 194, 202, 203, 211, 212, 218, 220, 221, 223, 229

Offset: 1

Gerasimov Sergey, Oct 14 2013

Primes dividing k^2 + 2 at least twice are in A033200. - Charles R Greathouse IV, Oct 14 2013

			4 is in the sequence because 4^2 + 2 = 18 = 2 * 3^2, which is not squarefree.
5 is in the sequence because 5^2 + 2 = 27 = 3^3, which is not squarefree.
6 is not in the sequence because 6^2 + 2 = 38 = 2 * 19, which is squarefree.

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

Cf. A033200, A013929, A228140.

Select[Range[300], ! SquareFreeQ[#^2 + 2] &] (* T. D. Noe, Oct 14 2013 *)
(* The following works in Mathematica versions prior to 6.0 *) Select[Range[250], MoebiusMu[#^2 + 2] == 0 &] (* Alonso del Arte, Oct 14 2013 *)

is(n)=!issquarefree(n^2+2) \\ Charles R Greathouse IV, Oct 14 2013

{k: k^2 + 2 is in A013929}.

cp's OEIS Frontend

A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

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A048861 a(n) = n^n - 1.

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A003628 Primes congruent to {5, 7} mod 8.

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A163183 Primes dividing 2^j + 1 for some odd j.

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

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A047471 Numbers that are congruent to {1, 3} mod 8.

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