cp's OEIS Frontend

A363414 a(n) = (1/2) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-4).

%I A363414 #16 Jun 10 2023 05:28:11
%S A363414 0,1,3,-18,-190,1035,25305,-120260,-5954940,22115925,2197084175,
%T A363414 -5141457750,-1173207584250,769657081375,856957094209125,
%U A363414 1127788828491000,-821262134429035000,-2922085673288364375,1000078365473764126875,6056214264965246443750,-1508740652939902034493750
%N A363414 a(n) = (1/2) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-4).
%C A363414 Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
%C A363414 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 A363414 Type 1: primes p that do not divide any element of the sequence {a(n)}.
%C A363414 In this case, unlike in A105750, the set of type 1 primes is conjecturally empty; it appears that every prime p divides some term of this sequence.
%C A363414 Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
%C A363414 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.
%C A363414 Moll's conjecture 5.5 extends to this sequence: for the primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
%C A363414 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 A363414 We conjecture that the set of type 3 primes consists of primes p == 3 (mod 4), equivalently, rational 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.
%H A363414 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 A363414 a(n) = Sum_{k = 0..floor(n/2)} (-4)^k*Stirling1(n+1,n-2*k).
%F A363414 P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) with
%F A363414 a(0) = 0 and a(1) = 1.
%e A363414 Type 2 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 1..100] = [0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 14, 12, 13, 12, 12, 14, 13, 14, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 19, 18, 19, 18, 18, 21, 19, 20, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25].
%e A363414 Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
%e A363414 Type 3 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 1..100] = [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.
%p A363414 a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
%p A363414 (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) )/(n - 1) end if; end:
%p A363414 seq(a(n), n = 0..20);
%Y A363414 Cf. A105750, A105751, A363409 - A363416.
%K A363414 sign,easy
%O A363414 0,3
%A A363414 _Peter Bala_, Jun 01 2023