cp's OEIS Frontend

A363412 a(n) = 1/sqrt(3) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-3).

%I A363412 #19 Mar 27 2025 04:42:07
%S A363412 0,1,3,-12,-140,420,13692,-23744,-2366784,126000,641927440,1306329024,
%T A363412 -252172135488,-1138135788608,135593735484480,999117715814400,
%U A363412 -95707279587325952,-1013737882826462976,85873512374909086464,1217682899871358735360,-95486742904897158097920
%N A363412 a(n) = 1/sqrt(3) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-3).
%C A363412 Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
%C A363412 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 A363412 Type 1: primes p that do not divide any element of the sequence {a(n)}.
%C A363412 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 A363412 Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
%C A363412 We conjecture that the set of type 2 primes consists of primes of the form p == 1 (mod 6), i.e., rational primes that split in the field extension Q(sqrt(-3)) of Q, together with the prime p = 2. See A002476.
%C A363412 Moll's conjecture 5.5 extends to this sequence and takes the form:
%C A363412 (i) the 2-adic valuation v_2(a(n)) ~ n/2 as n -> oo.
%C A363412 (ii) for prime p == 1 (mod 6), the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
%C A363412 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 A363412 We conjecture that the set of type 3 primes consists of primes of the form p == 5 (mod 6), i.e., the rational primes that remain inert in the field extension Q(sqrt(-3)) of Q, together with the prime p = 3, which ramifies in Q(sqrt(-3)). See A007528.
%H A363412 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 A363412 a(n) = Sum_{k = 0..floor((n+1)/2)} (-3)^k*Stirling1(n+1,n-2*k).
%F A363412 P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) with a(0) = 0 and a(1) = 1.
%F A363412 Conjectures: 3 does not divide a(3*n+1) for all n; the 3-adic valuation v_3(a(3*n)) = v_3(a(3*n-1)) for all n.
%e A363412 Type 2 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 1..60] = [0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 11, 10].
%e A363412 Note that v_7(a(60)) = 10 = 60/(7 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
%e A363412 Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 0, 1, 1, 0, 0, 0, 3, 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, 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 A363412 a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
%p A363412 (2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) )/(n - 1) end if; end:
%p A363412 seq(a(n), n = 0..20);
%t A363412 Table[Im[Product[1 + k*Sqrt[-3], {k, 0, n}]] / Sqrt[3], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 27 2025 *)
%Y A363412 Cf. A105750, A105751, A363409 - A363416
%K A363412 sign,easy
%O A363412 0,3
%A A363412 _Peter Bala_, Jun 01 2023