A363413 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-4).

1, 1, -7, -43, 245, 4045, -20795, -729335, 3118985, 217496825, -667140175, -97338843875, 149451128125, 61156245509125, 18055448952125, -51399370203595375, -123577855227019375, 55722247285947360625, 266112415762709595625, -75739843360243364046875, -560236984557463079546875

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, 3, 11, 19, 23, 31, 47, 59, 67, 71, 79, ...}.

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 4), equivalently, rational primes that split in the field extension Q(sqrt(-1)) of Q. See A002144.

Moll's conjecture 5.5 extends to this sequence and takes the form: for primes 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 begins {7, 43, 83, 131, 163, 167, ...}.

Taken together, the type 1 and type 3 primes appear to consist of primes p == 3 (mod 4), equivalently, primes that remain inert in the field extension Q(sqrt(-1)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002145.

Examples

			Type 2 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 5, 3, 3, 3, 4, 4, 4, 6, 6, 5, 5, 5, 6, 6, 6, 6, 6, 8, 8, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 11, 12, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 15, 14, 14, 14, 15, 15, 15, 15, 15, 17, 19, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 21, 20, 20, 20, 21, 21, 21, 21, 21, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 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..100] = [0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1], 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*(4*n^2 - 8*n + 5)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Formula

a(n) = Sum_{k = 0..floor((n+1)/2)} (-4)^k*Stirling1(n+1,n+1-2*k).

a(n+1)/a(n) = 1 - (2*n + 2)*tan( Sum_{k = 1..n} arctan(2*k) ).

P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) with

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