A003628 -id:A003628 - cp's OEIS frontend

A002479 Numbers of the form x^2 + 2*y^2.

0, 1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 17, 18, 19, 22, 24, 25, 27, 32, 33, 34, 36, 38, 41, 43, 44, 48, 49, 50, 51, 54, 57, 59, 64, 66, 67, 68, 72, 73, 75, 76, 81, 82, 83, 86, 88, 89, 96, 97, 98, 99, 100, 102, 107, 108, 113, 114, 118, 121, 123, 128, 129, 131

Offset: 1

N. J. A. Sloane

A positive number k belongs to this sequence if and only if every prime p == 5, 7 (mod 8) dividing k occurs to an even power. - Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002
Norms of numbers in Z[sqrt(-2)]. - Alonso del Arte, Sep 23 2014
Euler (E256) shows that these numbers are closed under multiplication, according to the Euler Archive. - Charles R Greathouse IV, Jun 16 2016
In addition to the previous comment: The proof was already given 1100 years before Euler by Brahmagupta's identity (a^2 + m*b^2)*(c^2 + m*d^2) = (a*c - m*b*d)^2 + m*(a*d + b*c)^2. - Klaus Purath, Oct 07 2023

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 421.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..3148 (first 1000 terms from T. D. Noe)
L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
L. Euler, (E256) Specimen de usu observationum in mathesi pura, Novi Commentarii academiae scientiarum Petropolitanae 6 (1761), pp. 185-230.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Complement of A097700. Subsequence of A000408. For primes see A033203.

Cf. A035251, A027748, A124010, A003628.

a002479 n = a002479_list !! (n-1)
a002479_list = 0 : filter f [1..] where
   f x = all (even . snd) $ filter ((`elem` [5,7]) . (`mod` 8) . fst) $
                            zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Feb 20 2014

[n: n in [0..131] | NormEquation(2, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

lis:={}; M:=50; M2:=M^2;
for x from 0 to M do for y from 0 to M do
if x^2+2*y^2 <= M2 then lis:={op(lis),x^2+2*y^2}; fi; od: od:
sort(convert(lis,list)); # N. J. A. Sloane, Apr 30 2015

q = 16; imax = q^2; Select[Union[Flatten[Table[x^2 + 2y^2, {y, 0, q/Sqrt[2]}, {x, 0, q}]]], # <= imax &] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
Union[#[[1]]+2#[[2]]&/@Tuples[Range[0,10]^2,2]] (* Harvey P. Dale, Nov 24 2014 *)

is(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,1]%8>4 && f[i,2]%2, return(0)));1 \\ Charles R Greathouse IV, Nov 20 2012

list(lim)=my(v=List()); for(a=0,sqrtint(lim\=1), for(b=0,sqrtint((lim-a^2)\2), listput(v,a^2+2*b^2))); Set(v) \\ Charles R Greathouse IV, Jun 16 2016

from itertools import count, islice
from sympy import factorint
def A002479_gen(): # generator of terms
    return filter(lambda n:all(p & 7 < 5 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A002479_list = list(islice(A002479_gen(),30)) # Chai Wah Wu, Jun 27 2022

A014566 Sierpiński numbers of the first kind: n^n + 1.

Original entry on oeis.org

2, 2, 5, 28, 257, 3126, 46657, 823544, 16777217, 387420490, 10000000001, 285311670612, 8916100448257, 302875106592254, 11112006825558017, 437893890380859376, 18446744073709551617, 827240261886336764178, 39346408075296537575425, 1978419655660313589123980

Offset: 0

Eric W. Weisstein

Sierpiński primes of the form n^n + 1 are {2,5,257,...} = A121270. The prime p divides a((p-1)/2) for p = {5,7,13,23,29,31,37,47,53,61,71,...} = A003628 Primes congruent to {5, 7} mod 8. p^2 divides a((p-1)/2) for prime p = {29,37,3373,...}. - Alexander Adamchuk, Sep 11 2006

n divides a(n-1) for even n, or 2n divides a(2n-1). a(2n-1)/(2n) = A124899(n) = {1, 7, 521, 102943, 38742049, 23775972551, 21633936185161, 27368368148803711, 45957792327018709121, ...}. 2^n divides a(2^n-1). A014566[2^n - 1] / 2^n = A081216[2^n - 1] = A122000[n] = {1, 7, 102943, 27368368148803711, 533411691585101123706582594658103586126397951, ...}. p+1 divides a(p) for prime p. a(p)/(p+1) = A056852[n] = {7, 521, 102943, 23775972551, 21633936185161, ...}. p^2 divides a((p-1)/2) for prime p = {29, 37, 3373} = A121999(n). - Alexander Adamchuk, Nov 12 2006

Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Maohua Le, Primes in the sequences n^n+1 and n^n-1, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, pp. 156-157.
Paulo Ribenboim, The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 74, 1989.

M. F. Hasler, Table of n, a(n) for n = 0..100
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ. Hse., 1990, Problem 17.
Eric Weisstein's World of Mathematics, Sierpiński Number of the First Kind.

Cf. A000312, A048861, A121270, A003628, A122000, A081216, A056852, A121999, A124899, A134883.

a(0) = 2; for n>0 Table[n^n+1,{n,1,20}] (* Alexander Adamchuk, Sep 11 2006 *)

A014566[n]:=if n=0 then 2 else n^n+1$
makelist(A014566[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

A014566(n)=n^n+1 /* M. F. Hasler, Jan 21 2009 */

For n>0, resultant of x^n+1 and nx-1. - Ralf Stephan, Nov 20 2004
E.g.f.: exp(x) + 1/(1+LambertW(-x)). - Vaclav Kotesovec, Dec 20 2014
Sum_{n>=1} 1/a(n) = A134883. - Amiram Eldar, Nov 15 2020

More terms from Erich Friedman

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.

Original entry on oeis.org

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

A005968 Sum of cubes of first n Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 10, 37, 162, 674, 2871, 12132, 51436, 217811, 922780, 3908764, 16558101, 70140734, 297121734, 1258626537, 5331629710, 22585142414, 95672204155, 405273951280, 1716768021816, 7272346018247, 30806152127640, 130496954475672, 552793970116297, 2341672834801754

Offset: 0

N. J. A. Sloane

From Alexander Adamchuk, Aug 07 2006: (Start)
The only two prime terms are a(2) = 2 and a(4) = 37.
The prime p divides a(p-1) iff p is in A045468.
The prime p divides a((p-1)/2) iff p is in A047650.
3^4 divides a(p) iff p is in A003628.
3^5 divides a(p) for p = {37,53,109,181,197,269,397,431,541,...}.
3^6 divides a(p) for p = {109,541,...}.
3^7 divides a(p) for p = {557,...}. (End)

Art Benjamin, Timothy A. Carnes, and Benoit Cloitre, Recounting the Sums of Cubes of Fibonacci Numbers, Congressus Numerantium, Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications, (William Webb, ed.), Vol 194, pp. 45-51, 2009.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 14.
A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Art Benjamin, Timothy A. Carnes, and Benoit Cloitre, Counting the Sums of Cubes of Fibonacci Numbers.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
David Treeby, Hidden Formulas in Fibonacci Tilings, Fibonacci Quart. 54 (2016), no. 1, 23-30.
Index entries for linear recurrences with constant coefficients, signature (4,3,-9,2,1).

Partial sums of A056570.  Cf. A119284 (alternating sum).
Cf. A045468, A047650, A003628.
Sums of other powers: A000071, A001654, A005969, A098531, A098532, A098533, A128697.

[(1/10)*( Fibonacci(3*n+2)-(-1)^(n)*6*Fibonacci(n-1)+5 ): n in [0..30]]; // G. C. Greubel, Jan 17 2018

with(combinat): l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+fibonacci(i)^3; printf(`%d,`,l[i]) od: # James Sellers, May 29 2000
A005968:=(-1+2*z+z**2)/(z-1)/(z**2+4*z-1)/(z**2-z-1); # conjectured by Simon Plouffe in his 1992 dissertation

f[n_]:=(Fibonacci[n]*Fibonacci[n+1]^2+(-1)^(n-1)*Fibonacci[n-1]+1)/2;Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Nov 22 2010 *)
Accumulate[Fibonacci[Range[0,20]]^3]
CoefficientList[Series[x*(1-2*x-x^2)/((1-x)*(1+x-x^2)*(1-4*x-x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

a(n)=(fibonacci(n)*fibonacci(n+1)^2+(-1)^(n-1)*fibonacci(n-1)+1)/2

a(n)=(fibonacci(3*n+2)-(-1)^(n)*6*fibonacci(n-1)+5)/10

PARI
```
a(n)=sum(i=1,n,fibonacci(i)^3)
```

G.f.: x*(1-2*x-x^2)/((1-x)*(1+x-x^2)*(1-4*x-x^2)). - Ralf Stephan, Apr 23 2004
a(n) = (1/2)*(F(n)*F(n+1)^2 + (-1)^(n-1)*F(n-1) + 1). - Benoit Cloitre, Aug 06 2004
a(n) = Sum_{i=1..n} A000045(i)^3.
a(n) = (1/10)*(F(3*n+2) - (-1)^(n)*6*F(n-1) + 5). - Art Benjamin and Timothy A. Carnes
a(n+5) = 4*a(n+4) + 3*a(n+3) - 9*a(n+2) + 2*a(n+1) + a(n). - Benoit Cloitre, Sep 12 2004

More terms from James Sellers, May 29 2000

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

Original entry on oeis.org

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

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)

A047566 Numbers that are congruent to {4, 5, 6, 7} mod 8.

Original entry on oeis.org

4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 68, 69, 70, 71, 76, 77, 78, 79, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109

Offset: 1

N. J. A. Sloane

Numbers having a 1 in position 2 of their binary expansion. One of the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. - Jeremy Gardiner, Jan 22 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Maths Magic, Mystery Calculator.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A003628 (primes).

Mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421.

a047566 n = a047566_list !! (n-1)
a047566_list = [n | n <- [1..], mod n 8 > 3]
-- Reinhard Zumkeller, Dec 29 2012

A047566:= n-> n+3 + 4*iquo(n-1, 4):
seq(A047566(n), n=1..100);  # Alois P. Heinz, Aug 22 2011

Flatten[# + {4, 5, 6, 7}&/@(8Range[0, 14])] (* Harvey P. Dale, Feb 02 2011 *)

G.f.: x*(4+x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 19 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (4*n+1-(-1)^n-(-1)^((n+1)/2)-(-1)^(n/2)-(-1)^(-(n+1)/2)-(-1)^(-n/2))/2. (End)
E.g.f.: 1 + sin(x) - cos(x) + sinh(x) + 2*x*exp(x). - Ilya Gutkovskiy, May 20 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 - 3*log(2)/8. - Amiram Eldar, Dec 26 2021

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.

A045355 Primes congruent to {2, 5, 7} mod 8.

Original entry on oeis.org

2, 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

Inert rational primes in the field Q(sqrt(-2)). - Eyal Gruss, Nov 30 2022

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

Cf. A000040.

[ p: p in PrimesUpTo(1000) | p mod 8 in {2, 5, 7} ]; // Vincenzo Librandi, Aug 07 2012

Select[Prime[Range[800]],MemberQ[{2,5, 7},Mod[#,8]]&] (* Vincenzo Librandi, Aug 07 2012 *)

{2} UNION A003628. - R. J. Mathar, Sep 19 2012

A121999 Primes p such that p^2 divides Sierpinski number A014566((p-1)/2).

Original entry on oeis.org

29, 37, 3373

Offset: 1

Alexander Adamchuk, Sep 11 2006

Subsequence of A003628.

No other terms below 10^11. - Max Alekseyev, Sep 18 2010

Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind.

Cf. A014566, A003628.

Do[p=Prime[n];f=((p-1)/2)^((p-1)/2)+1;If[IntegerQ[f/p^2],Print[p]],{n,1,3373}]

{ forprime(p=3, 10^11, if(Mod((p-1)/2, p^2)^((p-1)/2)==-1, print(p); )) } \\ Max Alekseyev, Sep 18 2010

Elements of A125854 that are congruent to 5 or 7 modulo 8, i.e., primes p such that p == 5 or 7 (mod 8) and 2^(p-1) == 1+p (mod p^2). - Max Alekseyev, Sep 18 2010

cp's OEIS Frontend

A002479 Numbers of the form x^2 + 2*y^2.

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A014566 Sierpiński numbers of the first kind: n^n + 1.

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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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A005968 Sum of cubes of first n Fibonacci numbers.

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

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

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