A140434 -id:A140434 - cp's OEIS frontend

A018805 Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y)=1}.

1, 3, 7, 11, 19, 23, 35, 43, 55, 63, 83, 91, 115, 127, 143, 159, 191, 203, 239, 255, 279, 299, 343, 359, 399, 423, 459, 483, 539, 555, 615, 647, 687, 719, 767, 791, 863, 899, 947, 979, 1059, 1083, 1167, 1207, 1255, 1299, 1391, 1423, 1507, 1547, 1611, 1659, 1763

Offset: 1

David W. Wilson

Number of positive rational numbers of height at most n, where the height of p/q is max(p, q) when p and q are relatively prime positive integers. - Charles R Greathouse IV, Jul 05 2012
The number of ordered pairs (i,j) with 1<=i<=n, 1<=j<=n, gcd(i,j)=d is a(floor(n/d)). - N. J. A. Sloane, Jul 29 2012
Equals partial sums of A140434 (1, 2, 4, 4, 8, 4, 12, 8, ...) and row sums of triangle A143469. - Gary W. Adamson, Aug 17 2008
Number of distinct solutions to k*x+h=0, where 1 <= k,h <= n. - Giovanni Resta, Jan 08 2013
a(n) is the number of rational numbers which can be constructed from the set of integers between 1 and n, without combination of multiplication and division. a(3) = 7 because {1, 2, 3} can only create {1/3, 1/2, 2/3, 1, 3/2, 2, 3}. - Bernard Schott, Jul 07 2019

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 110-112.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954. See Theorem 332.

Olivier Gérard, Table of n, a(n) for n = 1..100000 [Replaces an earlier b-file from Charles R Greathouse IV]
Jin-Yi Cai and Eric Bach, On testing for zero polynomials by a set of points with bounded precision, Theoret. Comput. Sci. 296 (2003), no. 1, 15-25. MR1965515 (2004m:68279).
Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Eric Weisstein's World of Mathematics, Carefree Couple

Cf. A015614, A002088, A100613 (gcd > 1), A071778 (triples), A143469, A140434, A013661, A059956, A137243, A171503.
Cf. A177853 (partial sums).
The main diagonal of A331781, also of A333295.

a018805 n = length [()| x <- [1..n], y <- [1..n], gcd x y == 1]
-- Reinhard Zumkeller, Jan 21 2013

/* based on the first formula */ A018805:=func< n | 2*&+[ EulerPhi(k): k in [1..n] ]-1 >; [ A018805(n): n in [1..60] ]; // Klaus Brockhaus, Jan 27 2011

/* based on the second formula */ A018805:=func< n | n eq 1 select 1 else n^2-&+[ $$(n div j): j in [2..n] ] >; [ A018805(n): n in [1..60] ]; // Klaus Brockhaus, Feb 07 2011

N:= 1000; # to get the first N entries
P:= Array(1..N,numtheory:-phi);
A:= map(t -> 2*round(t)-1, Statistics:-CumulativeSum(P));
convert(A,list); # Robert Israel, Jul 16 2014

FoldList[ Plus, 1, 2 Array[ EulerPhi, 60, 2 ] ] (* Olivier Gérard, Aug 15 1997 *)
Accumulate[2*EulerPhi[Range[60]]]-1 (* Harvey P. Dale, Oct 21 2013 *)

PARI
```
a(n)=sum(k=1,n,moebius(k)*(n\k)^2)
```

A018805(n)=2 *sum(j=1, n, eulerphi(j)) - 1;
for(n=1, 99, print1(A018805(n), ", ")); /* show terms */

a(n)=my(s); forsquarefree(k=1,n, s+=moebius(k)*(n\k[1])^2); s \\ Charles R Greathouse IV, Jan 08 2018

from sympy import sieve
def A018805(n): return 2*sum(t for t in sieve.totientrange(1,n+1)) - 1 # Chai Wah Wu, Mar 23 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A018805(n): # based on second formula
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A018805(k1)
        j, k1 = j2, n//j2
    return n*(n-1)-c+j # Chai Wah Wu, Mar 24 2021

a(n) = 2*(Sum_{j=1..n} phi(j)) - 1.
a(n) = n^2 - Sum_{j=2..n} a(floor(n/j)).
a(n) = 2*A015614(n) + 1. - Reinhard Zumkeller, Apr 08 2006
a(n) = 2*A002088(n) - 1. - Hugo van der Sanden, Nov 22 2008
a(n) ~ (1/zeta(2)) * n^2 = (6/Pi^2) * n^2 as n goes to infinity (zeta is the Riemann zeta function, A013661, and the constant 6/Pi^2 is 0.607927..., A059956). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 18 2001
a(n) ~ 6*n^2/Pi^2 + O(n*log n). - N. J. A. Sloane, May 31 2020
a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^2. - Benoit Cloitre, May 11 2003
a(n) = A000290(n) - A100613(n) = A015614(n) + A002088(n). - Reinhard Zumkeller, Jan 21 2013
a(n) = A242114(floor(n/k),1), 1<=k<=n; particularly a(n) = A242114(n,1). - Reinhard Zumkeller, May 04 2014
a(n) = 2 * A005728(n) - 3. - David H Post, Dec 20 2016
a(n) ~ 6*n^2/Pi^2, cf. A059956. [Hardy and Wright] - M. F. Hasler, Jan 20 2017
G.f.: (1/(1 - x)) * (-x + 2 * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^2). - Ilya Gutkovskiy, Feb 14 2020

More terms from Reinhard Zumkeller, Apr 08 2006

Link to Moree's paper corrected by Peter Luschny, Aug 08 2009

A319998 a(n) = Sum_{d|n, d is even} mu(n/d)*d, where mu(n) is Moebius function A008683.

Original entry on oeis.org

0, 2, 0, 2, 0, 4, 0, 4, 0, 8, 0, 4, 0, 12, 0, 8, 0, 12, 0, 8, 0, 20, 0, 8, 0, 24, 0, 12, 0, 16, 0, 16, 0, 32, 0, 12, 0, 36, 0, 16, 0, 24, 0, 20, 0, 44, 0, 16, 0, 40, 0, 24, 0, 36, 0, 24, 0, 56, 0, 16, 0, 60, 0, 32, 0, 40, 0, 32, 0, 48, 0, 24, 0, 72, 0, 36, 0, 48, 0, 32, 0, 80, 0, 24, 0, 84, 0, 40, 0, 48, 0, 44, 0, 92, 0, 32, 0, 84, 0, 40, 0, 64, 0, 48, 0

Offset: 1

Antti Karttunen, Oct 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A008683, A059841, A140434, A146076, A319997.

Rest[CoefficientList[Series[Sum[2*MoebiusMu[k]*x^(2*k)/(1 - x^(2*k))^2, {k, 1, 100}], {x, 0, 100}], x]] (* Vaclav Kotesovec, Nov 03 2018 *)

A319998(n) = sumdiv(n,d,(!(d%2))*moebius(n/d)*d);

A319998(n) = if(n%2, 0, 2*eulerphi(n/2));

a(n) = Sum_{d|n} A059841(d)*A008683(n/d)*d.
a(n) = A000010(n) - A319997(n).
a(2n) = 2*A000010(n), a(2n+1) = 0.
G.f.: Sum_{k>=1} 2*mu(k)*x^(2*k)/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Nov 02 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/(2*Pi^2) = 0.151981... . - Amiram Eldar, Nov 12 2022

A353747 a(n) = phi(n) + A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

2, 2, 4, 3, 7, 4, 11, 5, 10, 7, 17, 6, 23, 11, 14, 9, 29, 10, 35, 11, 22, 17, 41, 10, 29, 23, 26, 17, 51, 14, 59, 17, 34, 29, 39, 16, 67, 35, 46, 19, 77, 22, 83, 27, 36, 41, 89, 18, 67, 29, 58, 35, 99, 26, 61, 29, 70, 51, 111, 22, 119, 59, 56, 33, 81, 34, 127, 45, 82, 39, 137, 28, 143, 67, 58, 53, 95, 46, 151, 35, 70

Offset: 1

Antti Karttunen, May 06 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A064989, A104141, A140434, A151799, A353748.

f1[p_, e_] := (p - 1)*p^(e - 1); f2[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 80] (* Amiram Eldar, May 07 2022 *)

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353747(n) = (eulerphi(n)+A064989(n));

a(n) = A000010(n) + A064989(n).
For n >= 0, a(4n+2) = a(2n+1).
For n > 1, a(n) = A140434(n) - A353748(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) + 3/Pi^2 = 0.524667479..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A353748 a(n) = phi(n) - A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 3, 2, 1, 3, 2, 1, 1, 2, 7, 3, 2, 1, 5, 2, 3, 3, 6, 11, 1, 10, 7, 5, 2, 1, 15, 6, 3, 9, 8, 5, 1, 2, 13, 3, 2, 1, 13, 12, 3, 3, 14, 17, 11, 6, 13, 5, 10, 19, 19, 2, 5, 5, 10, 1, 1, 16, 31, 15, 6, 5, 19, 6, 9, 3, 20, 1, 5, 22, 19, 25, 2, 5, 29, 38, 3, 3, 14, 25, 1, 10, 33, 5, 12, 17, 25, 2, 3, 21

Offset: 1

Antti Karttunen, May 06 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A064989, A140434, A151799, A353747.

f1[p_, e_] := (p - 1)*p^(e - 1); f2[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, May 07 2022 *)

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353748(n) = (eulerphi(n)-A064989(n));

a(n) = A000010(n) - A064989(n).
For n >= 0, a(4n+2) = a(2n+1).
For n > 1, a(n) = A140434(n) - A353747(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/Pi^2 - (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.0832596219... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A344597 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^4 - floor((n-1)/k)^4).

Original entry on oeis.org

1, 14, 64, 160, 368, 592, 1104, 1520, 2400, 3056, 4640, 5264, 7824, 8736, 11776, 13216, 17984, 18384, 25344, 26080, 33312, 35120, 45584, 44320, 58480, 58512, 72000, 73200, 92624, 86848, 113520, 110144, 132640, 132416, 162816, 152112, 194544, 185616, 220416

Offset: 1

Seiichi Manyama, May 24 2021

nonn

Cf. A000583, A082540, A140434, A344596.

a[n_] := Sum[MoebiusMu[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^4), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 24 2021 *)

a(n) = sum(k=1, n, moebius(k)*((n\k)^4-((n-1)\k)^4));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^4))

from functools import lru_cache
@lru_cache(maxsize=None)
def A082540(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082540(k1)
        j, k1 = j2, n//j2
    return n*(n**3-1)-c+j
def A344597(n): return A082540(n)-A082540(n-1) # Chai Wah Wu, May 09 2025

Sum_{k=1..n} a(k) * floor(n/k) = n^4.
Sum_{k=1..n} a(k) = A082540(n).
G.f.: Sum_{k >= 1} mu(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.

A086227 a(n) = Sum_{1<=k<=4n, gcd(k,n)=1} (i^ktan(kPi/(4n)))/(4*i), where i is the imaginary unit.

Original entry on oeis.org

-1, 2, -2, 2, -4, 4, -4, 6, -4, 6, -8, 6, -8, 8, -8, 8, -12, 10, -8, 16, -12, 12, -16, 10, -12, 18, -16, 14, -16, 16, -16, 24, -16, 16, -24, 18, -20, 24, -16, 20, -32, 22, -24, 24, -24, 24, -32, 28, -20, 32, -24, 26, -36, 24, -32, 40, -28, 30, -32, 30, -32, 48, -32, 24, -48, 34, -32, 48, -32, 36, -48, 36, -36, 40, -40, 48, -48

Offset: 2

Benoit Cloitre, Aug 28 2003

sign

This seems to be (-1)^(n+1) times h(-4n^2) = (-1)^(n+1)*A000003(n^2), where h(k) is the class number. Verified for n <= 10^5. - Charles R Greathouse IV, Apr 28 2013

Amiram Eldar, Table of n, a(n) for n = 2..10000
Stanley Rabinowitz, Problem 2129, Crux Mathematicorum, Vol. 22, No. 3 (1996), p. 123; Solution to Problem 2129, by G. P. Henderson and Kurt Girstmair, ibid., Vol. 23, No. 4 (1997), pp. 246-249.

Cf. A000003, A204617,A079458, A140434.

f[p_, e_] := p^(e - 1) * Switch[Mod[p, 4], 2, 1, 1, p - 1, 3, p + 1]; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := If[EvenQ[n], -s[n], s[n]/2]; Array[a, 100, 2] (* Amiram Eldar, Mar 07 2022 *)

a(n)=round(real(1/4/I*sum(k=1,4*n,(I^k)*tan(Pi/4/n*if(gcd(k,n)-1,0,k)))))

a(n)=round(imag(sum(k=1,4*n,if(gcd(k,n)==1,I^k*tan(k*Pi/4/n))))/4) \\ Charles R Greathouse IV, Apr 25 2013

a(n)=my(s);for(k=1,2*n,if(gcd(2*k-1,n)==1,s-=(-1)^k*tan((2*k-1)*Pi/4/n))); round(s/4) \\ Charles R Greathouse IV, Apr 25 2013

a(n) = -A204617(n) if n is even, and A204617(n)/2 if n is odd (Rabinowitz, 1996). - Amiram Eldar, Mar 07 2022

a(n) = (-1)^(n+1)*A079458(n)/A140434(n). - Ridouane Oudra, Jun 23 2024

Definition corrected by Charles R Greathouse IV, Apr 25 2013

A143467 Triangle read by rows, A143315 * A128407, 1<=k<=n.

Original entry on oeis.org

1, 3, -1, 5, 0, -1, 7, -3, 0, 0, 9, 0, 0, 0, -1, 11, -5, -3, 0, 0, 1, 13, 0, 0, 0, 0, 0, -1, 15, -7, 0, 0, 0, 0, 0, 0, 17, 0, -5, 0, 0, 0, 0, 0, 0, 19, -9, 0, 0, -3, 0, 0, 0, 0, 1, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 23, -11, -7, 0, 0, 3, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Gary W. Adamson, Aug 17 2008

Row sums = A140434: (1, 2, 4, 4, 8, 4, 12, 8, 12,...).

Right border = mu(n), A008683: (1, -1, -1, 0, -1, 1,...).

			First few rows of the triangle =
1;
3, -1;
5, 0, -1;
7, -3, 0, 0;
9, 0, 0, 0, -1;
11, -5, -3, 0, 0, 1;
13, 0, 0, 0, 0, 0, -1;
...

Cf. A008683, A128407, A143315, A143467.

a(66) corrected by Georg Fischer, Aug 14 2023

A344596 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^3 - floor((n-1)/k)^3).

Original entry on oeis.org

1, 6, 18, 30, 60, 66, 126, 132, 198, 204, 330, 276, 468, 414, 552, 552, 816, 630, 1026, 840, 1116, 1050, 1518, 1128, 1740, 1476, 1890, 1692, 2436, 1704, 2790, 2256, 2820, 2544, 3384, 2556, 3996, 3186, 3960, 3408, 4920, 3420, 5418, 4260, 5112, 4686, 6486, 4560, 6930, 5340, 6816

Offset: 1

Seiichi Manyama, May 24 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Essentially 6*A102309 and 6*A326419.

Cf. A000578, A071778, A140434, A344597.

a[n_] := Sum[MoebiusMu[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^3), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 24 2021 *)

a(n) = sum(k=1, n, moebius(k)*((n\k)^3-((n-1)\k)^3));

a(n) = if(n<2, n, 3*sumdiv(n, d, moebius(n/d)*(d-1)*d));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3))

my(N=66, x='x+O('x^N)); Vec(x+6*sum(k=1, N, moebius(k)*x^(2*k)/(1-x^k)^3))

from sympy import mobius, divisors
def A344596(n): return 3*sum(mobius(n//d)*d*(d-1) for d in divisors(n,generator=True)) if n>1 else 1 # Chai Wah Wu, May 09 2025

Sum_{k=1..n} a(k) * floor(n/k) = n^3.
Sum_{k=1..n} a(k) = A071778(n).
a(n) = 3 * Sum_{d|n} mu(n/d) * (d-1) * d for n > 1.
G.f.: Sum_{k >= 1} mu(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.
G.f.: x + 6 * Sum_{k>=1} mu(k) * x^(2*k)/(1 - x^k)^3.
a(2^k) = 3*2^(k-2)*(3*2^k-2) for k>0. - Chai Wah Wu, May 10 2025

A140435 Number of new lattice points created at each step in an n X n grid that are not visible.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 7, 5, 11, 1, 15, 1, 15, 13, 15, 1, 23, 1, 23, 17, 23, 1, 31, 9, 27, 17, 31, 1, 43, 1, 31, 25, 35, 21, 47, 1, 39, 29, 47, 1, 59, 1, 47, 41, 47, 1, 63, 13, 59, 37, 55, 1, 71, 29, 63, 41, 59, 1, 87, 1, 63, 53, 63, 33, 91, 1, 71, 49, 91, 1, 95, 1, 75, 69, 79, 33, 107, 1, 95, 53

Offset: 1

Gregg Whisler, Jun 25 2008

nonn

Cf. A018805, A100613 and A140434.

g[n_] := Table[ #^2 &[m], {m, 1, n + 1}] - FoldList[Plus, 1, 2 Array[EulerPhi, n, 2]] - Most[Flatten[ Append[{0}, Table[ #^2 &[m], {m, 1, n + 1}] - FoldList[Plus, 1, 2 Array[EulerPhi, n, 2]]]]]; g[80]

G.f.: -Sum_{k>=2} mu(k) * x^k * (1 + x^k) / (1 - x^k)^2. - Ilya Gutkovskiy, Sep 14 2021

More terms from Robert G. Wilson v, Jan 17 2011

A143316 Triangle read by rows, A054525 * A143315, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 4, 0, 1, 4, 2, 0, 1, 8, 0, 0, 0, 1, 4, 4, 2, 0, 0, 1, 12, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 12, 0, 4, 0, 0, 0, 0, 0, 1, 8, 8, 0, 0, 2, 0, 0, 0, 0, 1, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 4, 4, 4, 0, 2, 0, 0, 0, 0, 0, 1, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 12, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = 2*n-1.

Left border = A140434: (1, 2, 4, 4, 8, 4, 12, 8,...).

			First few rows of the triangle =
1;
2, 1;
4, 0, 1;
4, 2, 0, 1;
8, 0, 0, 0, 1;
4, 4, 2, 0, 0, 1;
12, 0, 0, 0, 0, 0, 1;
8, 4, 0, 2, 0, 0, 0, 1;
...

Cf. A054525, A140434, A143315.

a(19) corrected and more terms from Georg Fischer, Aug 14 2023

cp's OEIS Frontend

A018805 Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y)=1}.

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A319998 a(n) = Sum_{d|n, d is even} mu(n/d)*d, where mu(n) is Moebius function A008683.

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A353747 a(n) = phi(n) + A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

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A353748 a(n) = phi(n) - A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

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A344597 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^4 - floor((n-1)/k)^4).

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A086227 a(n) = Sum_{1<=k<=4*n, gcd(k,n)=1} (i^k*tan(k*Pi/(4*n)))/(4*i), where i is the imaginary unit.

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A143467 Triangle read by rows, A143315 * A128407, 1<=k<=n.

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A086227 a(n) = Sum_{1<=k<=4n, gcd(k,n)=1} (i^ktan(kPi/(4n)))/(4*i), where i is the imaginary unit.