OFFSET
0,3
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 {5, 11, 13, 31, 53, 79, 97, 113, 137, 157, 179, 193, 197, ...}.
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 begins {2, 3, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, ...} and consists of primes p == 1, 3, 7 or 9 (mod 20), equivalently, rational primes that split in the field extension Q(sqrt(-5)) of Q, together with the prime p = 2. See A139513.
It can be shown that the 2-adic valuation v_2(a(n)) = floor((n+1)/4).
Moll's conjecture 5.5 extends to this sequence: for type two primes p > 2, 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 {17, 19, 37, 59, 71, 73, 131, 151, 173, 191, 199, ...}.
Taken together, the type 1 and type 3 primes appear to consist of primes p == 11, 13, 17 or 19 (mod 20), equivalently, primes that remain inert in the field extension Q(sqrt(-5)) of Q, together with the prime p = 5, which ramifies in Q(sqrt(-5)). See A003626.
LINKS
Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.
FORMULA
a(n) = Sum_{k = 0..floor((n+1)/2)} (-5)^k*Stirling1(n+1,n+1-2*k).
a(n+1)/a(n) = 1 - (5*n + 5)*1/sqrt(5)*tan(Sum_{k = 1..n} arctan(sqrt(5)*k).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) with
a(0) = a(1) = 1.
EXAMPLE
Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 0..80] = [0, 0, 2, 3, 1, 3, 3, 3, 4, 4, 4, 5, 5, 5, 7, 7, 7, 8, 8, 8, 9, 9, 9, 12, 12, 12, 13, 13, 13, 14, 14, 14, 16, 16, 16, 17, 17, 17, 18, 18, 18, 20, 20, 20, 21, 21, 21, 22, 22, 22, 25, 25, 25, 26, 26, 26, 27, 27, 27, 29, 29, 29, 30, 30, 30, 31, 31, 31, 33, 33, 33, 34, 34, 34, 35, 35, 35, 39, 39, 40, 40].
Note that v_3(a(80)) = 40 = 80/(3 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 17: the sequence of 17-adic valuations [v_17(a(n)) : n = 0..100] = [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.
MAPLE
a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Jun 01 2023
STATUS
approved