cp's OEIS Frontend

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

%I A363410 #18 Jun 10 2023 07:33:31
%S A363410 0,1,3,-6,-90,45,5607,8316,-616572,-2517075,106354215,779869134,
%T A363410 -26562900078,-299503403199,9075456298755,144911485323000,
%U A363410 -4066415773786872,-87372799002303111,2313066895842715947,64609858869087786210,-1627745411473223627970
%N A363410 a(n)= 1/sqrt(2) * the imaginary part of  Product_{k = 1..n} (1 + k*sqrt(-2)).
%C A363410 Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
%C A363410 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.
%C A363410 Type 1: primes p that do not divide any element of the sequence {a(n)}.
%C A363410 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.
%C A363410 Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
%C A363410 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.
%C A363410 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.
%C A363410 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.
%C A363410 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.
%H A363410 Victor H. Moll, <a href="http://www.tulane.edu/~vhm/papers_html/xn-final.pdf">An arithmetic conjecture on a sequence of arctangent sums</a>, 2012, see f_n.
%F A363410 (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.
%F A363410 a(n) = Sum_{k = 0..floor((n+1)/2)} (-2)^k*Stirling1(n+1,n+1-2*k).
%e A363410 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].
%e A363410 Note that v_3(a(100)) = 50 = 100/(3 - 1), in agrement with the asymptotic growth for type 2 primes conjectured above.
%e A363410 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.
%p A363410 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:
%p A363410 seq(a(n), n = 0..20);
%Y A363410 Cf. A105750, A105751, A363409 - A363416.
%K A363410 sign,easy
%O A363410 0,3
%A A363410 _Peter Bala_, Jun 01 2023