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

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

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 {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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A105750, A363410 - 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*(2*n^2 - 4*n + 3)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Mathematica

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

Formula

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.