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Original name was: Primes of the form x^2 + 4*x*y - y^2.

Discriminant = 20. Class number = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a,b,c) = 1 (primitive).

Values of the quadratic form are {0, 1, 4} mod 5, so this is a subsequence of A038872. - R. J. Mathar, Jul 30 2008

Is this the same sequence as A038872? [Yes. See a comment in A038872, and the comment by Jianing Song below. - Wolfdieter Lang, Jun 19 2019]

Also primes of the form u^2 - 5v^2. The transformation {u,v}={x+2y,y} transforms it into the one in the title. - Tito Piezas III, Dec 28 2008

From Jianing Song, Sep 20 2018: (Start)

Yes, this is a duplicate of A038872. For primes p congruent to {1, 4} mod 5, they split in the ring Z[(1+sqrt(5))/2]. Since Z[(1+sqrt(5))/2] is a UFD, they are reducible in Z[(1+sqrt(5))/2], so we have p = e*((a + b*sqrt(5))/2)*((a - b*sqrt(5))/2), where a and b have the same parity and e = +-1. WLOG we can suppose e = 1, otherwise substitute a, b by (a+5*b)/2 and (a+b)/2. Now we show that there exists integer u, v such that p = (u + v*sqrt(5))*(u - v*sqrt(5)) = u^2 - 5*v^2.

(i) If u, v are both even, then choose u = a/2, v = b/2.

(ii) If u, v are both odd, 4 | (a-b), then choose u = (3*a+5*b)/4, v = (3*b+a)/4.

(iii) If u, v are both odd, 4 | (a+b), then choose u = (3*a-5*b)/4, v = (3*b-a)/4.

Hence every prime congruent to {1, 4} mod 5 is of the form u^2 - 5*v^2. On the other hand, u^2 - 5*v^2 == 0, 1, 4 (mod 5). So these two sequences are the same.

Also primes of the form x^2 - x*y - y^2 (discriminant 5) with 0 <= x <= y (or x^2 + x*y - y^2 with x, y nonnegative). (End) [Comment revised by Jianing Song, Feb 24 2021]

Original name was: a(n) = the least number k such that cos(2pi/k) is an algebraic number of a prime(n)-smooth degree, but not prime(n-1)-smooth.

Comments from N. J. A. Sloane, Jan 07 2013: (Start)

This is a duplicate of A066674. This follows from the following argument. The degree of the minimal polynomial of cos(2*Pi/k) is phi(k)/2, where phi is Euler's totient function. Then a(n) is the least number k such that prime(n) is the largest prime dividing phi(k) and prime(n-1) does not divide phi(k)/2. For the rest of the proof see Bjorn Poonen's remarks in A066674.

It also seems likely that this is the same as A035095, but this is an open problem.

Conjecture: this sequence contains only primes (this would follow if this is indeed the same as A035095).

(End)

If there is a set of consecutive integers in this sequence starting at k, this means that k-1 is a good approximation to Pi.

If the set of successive integers is longer that approximation k-1 better (see A138336). [Sentence is not clear - N. J. A. Sloane, Dec 09 2017]

Comment from Joerg Arndt, Mar 17 2008: Does Mathematica's N[((quantity)), n] round a number (if so, to what base?) or truncate it? Is Mathematica's Recognize[] guaranteed to give the correct relation? I do not think so: that would be a major breakthrough. That is, this sequence may not even be well-defined.

This sequence is indeed ill defined. One can get the same approximation of Pi to a given precision with infinitely many distinct quadratic polynomials and any such polynomial that gives Pi to n+1 digits also gives Pi to n digits, so this sequence shouldn't have any term. Also, the 18-digit "root" given in the example isn't a root, the polynomial has a value of -5e-13 at this x-value. - M. F. Hasler, May 21 2025

Previous name was: a(n) is the smallest integer greater than a(n-1) such that a(1)*a(2)*...*a(n) + 1 is prime.

From Sean A. Irvine, Dec 05 2024: (Start)

For better versions of this sequence see A216993 and A378723.

This sequence is retained because of the terms given in the Brown link.

There can be more the one solution with the same smallest maximum denominator. For example, if n=8, we have:

1/3 + 1/5 + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20 = 1,

1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/15 + 1/18 + 1/20 = 1.

The definition of this sequence does not specify which of these should be retained and various rows given here are not consistent in their selection. In A378723, the second solution is taken because 10 < 12 when reading the denominators from the right. In A216993, the first solution is taken because 3 < 4 when reading the denominators from the left. (End)

Such number digits after decimal point of number Pi where the approximation to the number Pi by a root of a polynomial of 2 degree does not improve the accuracy exactly n next digits.

It seems that sequence A138335, and therefore also this sequence, are ill defined. - M. F. Hasler, May 21 2025

Name was: Primes of the form x^2 + 3*x*y - 3*y^2 (as well as of the form x^2 + 5*x*y + y^2).

Discriminant = 21. Class number = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac and gcd(a,b,c)=1 (primitive).

Primes of the form 6n+1 which cannot be expressed as 7k-1, 7k-2, or 7k-4. a(n)^2 == 1 (mod 24). - Gary Detlefs, Jan 26 2014

Besides 7 (which divides 21), primes of the form p == 1 (mod 3) and either == 1 or 2 or 4 (mod 7). For the other class, the primes represented by the principal form [3, 3, -1] (or primitive forms equivalent to this) are besides 3 (which divides 21), congruent to 2 (mod 3) and also to 3, 5, 6 (mod 7). For the primes of both classes see A038893. - Wolfdieter Lang, Jun 19 2019

For precise definition see Knuth (1997).

It appears that there might be a missing number (26) in this data list. Only when the sequence is 1, 1, 2, 6, 26, 158, 1330, 15414, 245578, 5382862 does it then match with A323842 via exp(t)*EGF(A323842). - Michael D. Weiner, Sep 23 2019

For correct version of this sequence see A135922. - Alois P. Heinz, Sep 24 2019