A363412 a(n) = 1/sqrt(3) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-3).

0, 1, 3, -12, -140, 420, 13692, -23744, -2366784, 126000, 641927440, 1306329024, -252172135488, -1138135788608, 135593735484480, 999117715814400, -95707279587325952, -1013737882826462976, 85873512374909086464, 1217682899871358735360, -95486742904897158097920

Offset: 0

Author

Peter Bala, Jun 01 2023

Keywords

• sign
• easy

Comments

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).

Moll (2012) studied the prime divisors of the terms of A105750 - the real part of Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.

Type 1: primes p that do not divide any element of the sequence {a(n)}.

We conjecture that in this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.

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 of the form 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 consists of primes of the form p == 5 (mod 6), i.e., 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 = 1..60] = [0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 11, 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, 0, 1, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2], 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, A105751, A363409 - A363416

Programs

Maple

a := proc(n) option remember; if n = 0 then 0 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[Im[Product[1 + k*Sqrt[-3], {k, 0, n}]] / Sqrt[3], {n, 0, 20}] (* Vaclav Kotesovec, Mar 27 2025 *)

Formula

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

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) = 0 and a(1) = 1.

Conjectures: 3 does not divide a(3*n+1) for all n; the 3-adic valuation v_3(a(3*n)) = v_3(a(3*n-1)) for all n.