cp's OEIS Frontend

A363416 a(n) = 1/sqrt(5) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-5).

%I A363416 #16 Jun 10 2023 05:17:30
%S A363416 0,1,3,-24,-240,1890,40446,-311472,-12038544,86898420,5614173180,
%T A363416 -36099955584,-3786960576672,20307572439336,3492389655843480,
%U A363416 -14110473458954880,-4223754447793582464,10493742733654512528,6488421280167604253616,-4618066393756887442560,-12344309538368967592151040
%N A363416 a(n) = 1/sqrt(5) * the imaginary  part of Product_{k = 0..n} 1 + k*sqrt(-5).
%C A363416 Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
%C A363416 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 A363416 Type 1: primes p that do not divide any element of the sequence {a(n)}.
%C A363416 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 A363416 Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
%C A363416 We conjecture that the set of type 2 primes 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.
%C A363416 Moll's conjecture 5.5 extends to this sequence and takes the form:
%C A363416 (i) the 2-adic valuation v_2(a(n)) ~ n/4 as n -> oo.
%C A363416 (ii) for the other primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
%C A363416 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 A363416 We conjecture that the set of type 3 primes consists 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.
%H A363416 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 A363416 P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) with
%F A363416 a(0) = 0 and a(1) = 1.
%F A363416 a(n) = Sum_{k = 0..floor((n+1)/2)} (-5)^k*Stirling1(n+1,n-2*k).
%e A363416 Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 1..80] = [0, 1, 1, 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, 41, 40, 40].
%e A363416 Note that v_3(a(80)) = 40 = 80/(3 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
%e A363416 Type 3 prime p = 11: the sequence of 11-adic valuations [v_11(a(n)) : n = 1..121] = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2], showing the oscillatory behavior for type 3 primes conjectured above.
%p A363416 a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
%p A363416 (2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) )/(n - 1) end if; end:
%p A363416 seq(a(n), n = 0..20);
%Y A363416 Cf. A105750, A105751, A363409 - A363415
%K A363416 sign,easy
%O A363416 0,3
%A A363416 _Peter Bala_, Jun 01 2023