A363411 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-3).

1, 1, -5, -32, 112, 2212, -5348, -292880, 276976, 64180144, 60400144, -21123205376, -68151050240, 9766562233792, 57568265355328, -6044149831446272, -54001800190537472, 4827069458763086080, 59568915131392086784, -4835221290238425841664, -77896195282519949963264

Offset: 0

Author

Peter Bala, Jun 01 2023

Keywords

• sign
• easy

Comments

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 {3, 11, 17, 23, 29, 41, 47, 53, 59, 83, 89, 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 (mod 6), i.e., rational primes that split in the field extension Q(sqrt(-3)) of Q, together with the prime p = 2. See A002476.

Moll's conjecture 5.5 extends to this sequence and takes the form:

(i) the 2-adic valuation v_2(a(n)) ~ n/2 as n -> oo.

(ii) for prime p == 1 (mod 6), 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 {5, 71, 179, 191, 227, 257, 263, ...}.

Taken together, the type 1 and type 3 primes appear to consist of all primes p == 5 (mod 6), that is, the rational primes that remain inert in the field extension Q(sqrt(-3)) of Q, together with the prime p = 3, which ramifies in Q(sqrt(-3)). See A007528.

Examples

			Type 2 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 0..60] = [0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 6, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 9, 8, 8, 9, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10].
Note that v_7(a(60)) = 10 = 60/(7 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.
		

Links

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

Crossrefs

Cf. A105750, A363409 - A363416.

Programs

Maple

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Mathematica

Table[Re[Product[1 + k*Sqrt[-3], {k, 0, n}]], {n, 0, 20}] (* Vaclav Kotesovec, Mar 27 2025 *)

Formula

a(n) = Sum_{k = 0..floor((n+1)/2)} (-3)^k*Stirling1(n+1,n+1-2*k).

a(n+1)/a(n) = 1 - (3*n + 3)*tan(Sum_{k = 1..n} arctan(sqrt(3)*k))/sqrt(3).

P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) with

a(0) = a(1) = 1.

Conjecture: the 5-adic valuation v_5(a(n+2)) = A079998(n) (checked up to n =

5000).