A363416 -id:A363416 - cp's OEIS frontend

A105750 Real part of Product_{k = 0..n} (1 + k*i), i = sqrt(-1).

1, 1, -1, -10, -10, 190, 730, -6620, -55900, 365300, 5864300, -28269800, -839594600, 2691559000, 159300557000, -238131478000, -38894192662000, -15194495654000, 11911522255750000, 29697351895900000, -4477959179352100000, -21683886333440500000, 2029107997508660900000

Offset: 0

Paul Barry, Apr 18 2005

Define u(n) as in A220448 and set f(n) = u(n)*f(n-1) for n >= 2, with f(1)=1 (this defines A220449). Then a(0)=1; a(n) = (-1)^(n+1)*f(n) for n >= 1. - N. J. A. Sloane, Dec 22 2012
From Peter Bala, Jun 03 2023: (Start)
Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes.
Type 1: primes p that do not divide any element of the sequence {a(n)}. The first few type 1 primes appear to be {3, 7, 11, 23, 31, 47, 59}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. The first few type 2 primes appear to be {2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97}.
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, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002144.
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).
We conjecture that the sets of type 1 and type 3 primes taken together consist of primes p == 3 (mod 4), equivalently, rational primes that remain inert in the field extension Q(sqrt(-1)) of Q. See A002145. (End)

N. J. A. Sloane, Table of n, a(n) for n = 0..100
V. H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A003703, A009454, A105751, A220448, A220449, A363409 - A363416.

A105750 := proc(n)
    mul(1-k*I,k=0..n) ;
    Re(%) ;
end proc: # R. J. Mathar, Jan 04 2013

x[n_] := x[n] = If[n == 1, 1, (x[n-1]+n)/(1-n*x[n-1])];
u[n_] := n*x[n-1]-1;
f[n_] := f[n] = If[n == 1, 1, u[n]*f[n-1]];
a[n_] := If[n == 0, 1, (-1)^(n+1)*f[n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 17 2023, after N. J. A. Sloane *)

from sympy.functions.combinatorial.numbers import stirling
def A105750(n): return sum(stirling(n+1,n+1-(k<<1),kind=1)*(-1 if k&1 else 1) for k in range((n+1>>1)+1)) # Chai Wah Wu, Feb 22 2024

a(n) = Re( Product_{k = 0..n} (1 - k*i) ).
Conjecture: a(n) -3*a(n-1) +(n^2-n+3)*a(n-2) +(-n^2+4*n-5)*a(n-3)=0. - R. J. Mathar, May 23 2014
From Peter Bala, May 28 2023: (Start)
a(n) = Sum_{k = 0..floor((n + 1)/2)} (-1)^k*|Stirling1(n+1, n-2*k+1)|, where Stirling1(n, k) = A048994(n,k).
(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(n^2 - 2*n + 2)*a(n-2) with a(0) = a(1) = 1 (see Moll, equation 1.16). Mathar's third-order recurrence above follows easily from this.
a(2*n) = (-1)^n*A009454(2*n+1) for n >= 0.
a(2*n-1) = (-1)^n*A003703(2*n) for n >= 1. (End)

Corrected by N. J. A. Sloane, Nov 05 2005

A105751 Imaginary part of Product_{k=0..n} (1 + k*i), i = sqrt(-1).

Original entry on oeis.org

0, 1, 3, 0, -40, -90, 1050, 6160, -46800, -549900, 3103100, 67610400, -271627200, -11186357000, 26495469000, 2416003824000, -1394099824000, -662595375078000, -936096296850000, 225382826562400000, 819329864480400000, -93217812901913700000, -570263312237604700000

Offset: 0

Paul Barry, Apr 18 2005

From Peter Bala, Jun 01 2023: (Start)
Compare with A105750(n) = the real part of Product_{k = 0..n} (1 + k*sqrt(-1)). 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 holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
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.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
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, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002144.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/4 as n -> oo.
(ii) for the other primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
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.
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. See A002145. (End)

			From _Peter Bala_, Jun 01 2023: (Start)
The sequence of 5-adic valuations [v_5(a(n)) : n = 4..100] = [1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 12, 11, 11, 13, 11, 12, 13, 13, 12, 12, 14, 13, 13, 14, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 18, 18, 18, 18, 18, 20, 19, 19, 20, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 24, 25, 25, 24, 24, 25, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior conjectured above.
The sequence of 3-adic valuations [v_3(a(n)) : n >= 4] begins [0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 3, 1, 0, 3, 3, 0, 1, 3, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 3, ...], exhibiting the oscillatory behavior for type 3 primes conjectured above. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A003703, A009454, A048994, A105750, A164652, A231531, A363409 - A363416.

a:= proc(n) option remember; `if`(n<2, n,
      ((2*n-1)*a(n-1)-(n^2-2*n+2)*n*a(n-2))/(n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 11 2018

Table[Im[Product[1+k*I,{k,0,n}]],{n,0,22}] (* James C. McMahon, Jan 27 2024 *)

a(n) = imag(prod(k=0, n, 1+k*I)); \\ Michel Marcus, Apr 11 2018

from sympy.functions.combinatorial.numbers import stirling
def A105751(n): return sum(stirling(n+1,n-(k<<1),kind=1)*(-1 if k&1 else 1) for k in range((n>>1)+1)) # Chai Wah Wu, Feb 22 2024

a(n) = ((2*n-1)*a(n-1)-(n^2-2*n+2)*n*a(n-2))/(n-1) for n > 1, a(n) = n for n < 2. - Alois P. Heinz, Apr 11 2018
From Peter Bala, May 27 2023:(Start)
a(n) = Sum_{k = 0..floor((n+1)/2)} (-1)^k*|Stirling1(n+1, n-2*k)|, where Stirling1(n, k) = A048994(n,k).
The triangular number n*(n+1)/2 divides a(n). See A164652. In particular, if p is an odd prime then p divides a(p).
a(2*n) = (-1)^(n+1)*A003703(2*n+1) for n >= 0.
a(2*n+1) = (-1)^(n+1)*A009454(2*n+2) for n >= 0. (End)

A363409 a(n) = the real part of Product_{k = 1..n} (1 + k*sqrt(-2)).

Original entry on oeis.org

1, 1, -3, -21, 27, 927, 387, -78111, -211167, 10887129, 61228629, -2278564101, -20995423317, 669639978711, 9055735268283, -263207953694367, -4900375484030367, 133357760824723281, 3278778524907635277, -84617763517115570709, -2669012118280627019109

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
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.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {2, 5, 13, 23, 29, 31, 47, 53, 61, 71, 101, ...}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 or 3 (mod 8), i.e., rational primes that split in the field extension Q(sqrt(-2)) of Q. See A033200.
Moll's conjecture 5.5 extends to this sequence and takes the form: for prime p == 1 or 3 (mod 8), the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
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.
We conjecture that the set of type 3 primes begins {7, 37, 79, 103, ...}.
Taken together, the type 1 and type 3 primes appear to consist of all primes p == 5 or 7 (mod 8), that is, the rational 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 A003628.

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 0..100] = [0, 0, 1, 1, 3, 2, 2, 3, 5, 5, 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, 37, 36, 36, 40, 37, 37, 38, 38, 38, 40, 40, 40, 41, 41, 41, 42, 42, 42, 47, 44, 44, 46, 45, 45, 46, 46, 46, 49, 49, 49, 50].
Note that v_3(a(100)) = 50 = 100/(3 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 0..101] = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1], showing the oscillatory behavior for type 3 primes conjectured above. It appears that v_7(a(7*n+3)) = 1 otherwise v_7(a(n)) = 0.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A363410 - A363416.

a := proc(n) option remember; if n = 0 then 1 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:
seq(a(n), n = 0..20);

Table[Re[Product[1+k*Sqrt[-2], {k, 0, n}]], {n, 0, 20}] (* James C. McMahon, Jan 28 2024 *)

a(n) = Sum_{k = 0..floor((n+1)/2)} (-2)^k*Stirling1(n+1, n+1-2*k).
a(n+1)/a(n) = 1 - (2*n + 2)*1/sqrt(2)*tan( Sum_{k = 1..n} arctan(sqrt(2)*k) ).
(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) with a(0) = 1 and a(1) = 1.

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

Original entry on oeis.org

0, 1, 3, -6, -90, 45, 5607, 8316, -616572, -2517075, 106354215, 779869134, -26562900078, -299503403199, 9075456298755, 144911485323000, -4066415773786872, -87372799002303111, 2313066895842715947, 64609858869087786210, -1627745411473223627970

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
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.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
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.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
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.
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.
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.
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.

			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].
Note that v_3(a(100)) = 50 = 100/(3 - 1), in agrement with the asymptotic growth for type 2 primes conjectured above.
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.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416.

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:
seq(a(n), n = 0..20);

(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.

a(n) = Sum_{k = 0..floor((n+1)/2)} (-2)^k*Stirling1(n+1,n+1-2*k).

A363415 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-5).

Original entry on oeis.org

1, 1, -9, -54, 426, 6426, -50274, -1465884, 10992996, 552727476, -3792193524, -312571718424, 1853425616616, 248005863100296, -1173524207653224, -263102748395914224, 865735128320476176, 359884863190774985616, -584551982838131141904, -616984573598760535235424, -155177934223071790979424

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
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.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {5, 11, 13, 31, 53, 79, 97, 113, 137, 157, 179, 193, 197, ...}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes begins {2, 3, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, ...} and 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.
It can be shown that the 2-adic valuation v_2(a(n)) = floor((n+1)/4).
Moll's conjecture 5.5 extends to this sequence: for type two primes p > 2, v_p(a(n)) ~ n/(p - 1) as n -> oo.
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.
We conjecture that the set of type 3 primes begins {17, 19, 37, 59, 71, 73, 131, 151, 173, 191, 199, ...}.
Taken together, the type 1 and type 3 primes appear to consist 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.

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 0..80] = [0, 0, 2, 3, 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, 40, 40].
Note that v_3(a(80)) = 40 = 80/(3 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 17: the sequence of 17-adic valuations [v_17(a(n)) : n = 0..100] = [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

a(n) = Sum_{k = 0..floor((n+1)/2)} (-5)^k*Stirling1(n+1,n+1-2*k).
a(n+1)/a(n) = 1 - (5*n + 5)*tan(Sum_{k = 1..n} arctan(sqrt(5)*k))/sqrt(5).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) with
a(0) = a(1) = 1.

A363411 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-3).

Original entry on oeis.org

1, 1, -5, -32, 112, 2212, -5348, -292880, 276976, 64180144, 60400144, -21123205376, -68151050240, 9766562233792, 57568265355328, -6044149831446272, -54001800190537472, 4827069458763086080, 59568915131392086784, -4835221290238425841664, -77896195282519949963264

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
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.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {3, 11, 17, 23, 29, 41, 47, 53, 59, 83, 89, 101, ...}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes 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.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/2 as n -> oo.
(ii) for prime p == 1 (mod 6), the p-adic valuation v_p(a(n)) ~ n/(p - 1)
as n -> oo.
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.
We conjecture that the set of type 3 primes begins {5, 71, 179, 191, 227, 257, 263, ...}.
Taken together, the type 1 and type 3 primes appear to consist of all primes p == 5 (mod 6), that is, 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.

			Type 2 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 0..60] = [0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 6, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 9, 8, 8, 9, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10].
Note that v_7(a(60)) = 10 = 60/(7 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Table[Re[Product[1 + k*Sqrt[-3], {k, 0, n}]], {n, 0, 20}] (* Vaclav Kotesovec, Mar 27 2025 *)

a(n) = Sum_{k = 0..floor((n+1)/2)} (-3)^k*Stirling1(n+1,n+1-2*k).
a(n+1)/a(n) = 1 - (3*n + 3)*tan(Sum_{k = 1..n} arctan(sqrt(3)*k))/sqrt(3).
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) = a(1) = 1.
Conjecture: the 5-adic valuation v_5(a(n+2)) = A079998(n) (checked up to n =
5000).

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

Original entry on oeis.org

0, 1, 3, -12, -140, 420, 13692, -23744, -2366784, 126000, 641927440, 1306329024, -252172135488, -1138135788608, 135593735484480, 999117715814400, -95707279587325952, -1013737882826462976, 85873512374909086464, 1217682899871358735360, -95486742904897158097920

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
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.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
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.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
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.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/2 as n -> oo.
(ii) for prime p == 1 (mod 6), the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
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.
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.

			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].
Note that v_7(a(60)) = 10 = 60/(7 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
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.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Table[Im[Product[1 + k*Sqrt[-3], {k, 0, n}]] / Sqrt[3], {n, 0, 20}] (* Vaclav Kotesovec, Mar 27 2025 *)

a(n) = Sum_{k = 0..floor((n+1)/2)} (-3)^k*Stirling1(n+1,n-2*k).
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.
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.

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

Original entry on oeis.org

1, 1, -7, -43, 245, 4045, -20795, -729335, 3118985, 217496825, -667140175, -97338843875, 149451128125, 61156245509125, 18055448952125, -51399370203595375, -123577855227019375, 55722247285947360625, 266112415762709595625, -75739843360243364046875, -560236984557463079546875

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
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.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {2, 3, 11, 19, 23, 31, 47, 59, 67, 71, 79, ...}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
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.
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.
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.
We conjecture that the set of type 3 primes begins {7, 43, 83, 131, 163, 167, ...}.
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.

			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].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
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.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

a(n) = Sum_{k = 0..floor((n+1)/2)} (-4)^k*Stirling1(n+1,n+1-2*k).
a(n+1)/a(n) = 1 - (2*n + 2)*tan( Sum_{k = 1..n} arctan(2*k) ).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) with
a(0) = a(1) = 1.

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

Original entry on oeis.org

0, 1, 3, -18, -190, 1035, 25305, -120260, -5954940, 22115925, 2197084175, -5141457750, -1173207584250, 769657081375, 856957094209125, 1127788828491000, -821262134429035000, -2922085673288364375, 1000078365473764126875, 6056214264965246443750, -1508740652939902034493750

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
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.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
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.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
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.
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.
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.
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.

			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].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
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.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

a(n) = Sum_{k = 0..floor(n/2)} (-4)^k*Stirling1(n+1,n-2*k).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) with
a(0) = 0 and a(1) = 1.

cp's OEIS Frontend

A105750 Real part of Product_{k = 0..n} (1 + k*i), i = sqrt(-1).

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A105751 Imaginary part of Product_{k=0..n} (1 + k*i), i = sqrt(-1).

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A363409 a(n) = the real part of Product_{k = 1..n} (1 + k*sqrt(-2)).

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

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A363415 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-5).

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A363411 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-3).

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

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