cp's OEIS Frontend

A363413 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-4).

%I A363413 #13 Jun 10 2023 05:27:19
%S A363413 1,1,-7,-43,245,4045,-20795,-729335,3118985,217496825,-667140175,
%T A363413 -97338843875,149451128125,61156245509125,18055448952125,
%U A363413 -51399370203595375,-123577855227019375,55722247285947360625,266112415762709595625,-75739843360243364046875,-560236984557463079546875
%N A363413 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-4).
%C A363413 Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
%C A363413 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.
%C A363413 Type 1: primes that do not divide any element of the sequence {a(n)}.
%C A363413 We conjecture that the set of type 1 primes begins {2, 3, 11, 19, 23, 31, 47, 59, 67, 71, 79, ...}.
%C A363413 Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
%C A363413 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 A363413 Moll's conjecture 5.5 extends to this sequence and takes the form: for primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
%C A363413 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 A363413 We conjecture that the set of type 3 primes begins {7, 43, 83, 131, 163, 167, ...}.
%C A363413 Taken together, the type 1 and type 3 primes appear to consist of primes p == 3 (mod 4), equivalently, 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 A363413 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 A363413 a(n) = Sum_{k = 0..floor((n+1)/2)} (-4)^k*Stirling1(n+1,n+1-2*k).
%F A363413 a(n+1)/a(n) = 1 - (2*n + 2)*tan( Sum_{k = 1..n} arctan(2*k) ).
%F A363413 P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) with
%F A363413 a(0) = a(1) = 1.
%e A363413 Type 2 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 5, 3, 3, 3, 4, 4, 4, 6, 6, 5, 5, 5, 6, 6, 6, 6, 6, 8, 8, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 11, 12, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 15, 14, 14, 14, 15, 15, 15, 15, 15, 17, 19, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 21, 20, 20, 20, 21, 21, 21, 21, 21, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25].
%e A363413 Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
%e A363413 Type 3 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 0..100] = [0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1], showing the oscillatory behavior for type 3 primes conjectured above.
%p A363413 a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
%p A363413 (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) )/(n - 1) end if; end:
%p A363413 seq(a(n), n = 0..20);
%Y A363413 Cf. A105750, A363409 - A363416.
%K A363413 sign,easy
%O A363413 0,3
%A A363413 _Peter Bala_, Jun 01 2023