A363410 a(n)= 1/sqrt(2) * the imaginary part of Product_{k = 1..n} (1 + k*sqrt(-2)).

0, 1, 3, -6, -90, 45, 5607, 8316, -616572, -2517075, 106354215, 779869134, -26562900078, -299503403199, 9075456298755, 144911485323000, -4066415773786872, -87372799002303111, 2313066895842715947, 64609858869087786210, -1627745411473223627970

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 p == 1 or 3 (mod 8), equivalently, rational primes that split in the field extension Q(sqrt(-2)) of Q. See A033200.

Moll's conjecture 5.5 extends to this sequence: for primes p of type 2, 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 p == 5 or 7 (mod 8), equivalently, 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 A033203.

Examples

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(n) : n = 1..100] =  [0, 1, 1, 2, 2, 2, 3, 4, 4, 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, 36, 36, 36, 37, 37, 37, 38, 38, 38, 41, 40, 40, 42, 41, 41, 42, 42, 42, 44, 44, 44, 45, 45, 45, 46, 46, 46, 49, 49, 49, 50].
Note that v_3(a(100)) = 50 = 100/(3 - 1), in agrement with the asymptotic growth for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(n) : n = 1..100] = [0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 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*(2*n^2 - 4*n + 3)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Formula

(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) with a(0) = 0 and a(1) = 1.

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