A079458 -id:A079458 - cp's OEIS frontend

A204617 Multiplicative with a(p^e) = p^(e-1)*H(p). H(2) = 1, H(p) = p - 1 if p == 1 (mod 4) and H(p) = p + 1 if p == 3 (mod 4).

1, 1, 4, 2, 4, 4, 8, 4, 12, 4, 12, 8, 12, 8, 16, 8, 16, 12, 20, 8, 32, 12, 24, 16, 20, 12, 36, 16, 28, 16, 32, 16, 48, 16, 32, 24, 36, 20, 48, 16, 40, 32, 44, 24, 48, 24, 48, 32, 56, 20, 64, 24, 52, 36, 48, 32, 80, 28, 60, 32, 60, 32, 96, 32, 48, 48, 68, 32

Offset: 1

José María Grau Ribas, Jan 17 2012

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

Cf. A000010, A072437.
Cf. A175647, A243381.
Cf. A079458, A062570.

with(numtheory):
a := n->add(jacobi(-1,d)*mobius(d)*n/d, d in divisors(n)):
seq(a(n), n = 1..60); # Peter Bala, Dec 26 2023

ar[p_,s_] := Which[Mod[p,4]==1, p^(s-1)*(p-1), Mod[p,4]==3, p^(s-1)*(p+1), True,p^(s-1)]; arit[1] = 1; arit[n_] := Product[ar[FactorInteger[n][[i,1]], FactorInteger[n][[i,2]]], {i, Length[FactorInteger[n]]}]; Array[arit, 100]

A204617(n) = { my(f=factor(n),p); prod(i=1, #f~, p=f[i, 1]; (p^(f[i, 2]-1)) * if(2==p,1,if(1==(p%4),p-1,p+1))); }; \\ Antti Karttunen, Nov 16 2021

a(n) = phi(n) if n is in A072437.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2) * Product_{primes p == 3 (mod 4)} (1 + 1/p^2) = 3*A243381/(8*A175647) = 0.409404... . - Amiram Eldar, Dec 24 2022
a(n) = n*Product_{primes p, p | n} (1 - A034947(p)/p) = Sum_{d | n} A034947(d)* mobius(d)*n/d. Cf. A000010(n) = Sum_{d | n} mobius(d)*n/d. - Peter Bala, Dec 26 2023
a(n) = A079458(n)/A062570(n). - Ridouane Oudra, Jun 04 2024

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

A218147 Degree of minimal polynomial satisfied by exp(8Piphi_2(1/n,1/n)), where phi_2 is defined in the Comments.

Original entry on oeis.org

2, 2, 4, 4, 12, 8, 18, 8, 30, 16, 36, 24, 32, 32, 64, 36, 90, 32, 96, 60, 132, 64, 100, 72, 162, 96, 196, 64, 240, 128, 240, 128, 192, 144, 324, 180, 288, 128, 400, 192, 462, 240, 288, 264, 552, 256, 588, 200, 512, 288, 676, 324, 480, 384, 720, 392, 870, 256

Offset: 3

Jason Kimberley, Oct 21 2012 and Apr 04 2016

Crandall defines phi_2(r_1,r_2) = (1/Pi^2) Sum_{positive & negative odd m_1, m_2} cos(Pi m_1 r_1) cos(Pi m_2 r_2) / (m_1^2+m_2^2).
Lemma: 4a(n) < n^2. Proof: 4a(2) = 2 < 2^2; 4a(4k+1) = 16k^2 < (4k+1)^2; 4a(4k+3) = (4k+2)(4k+4) = (4k+3)^2-1; 4a(p^2 k) = 4p^2 a(pk) < p^2(pk)^2 = (p^2 k)^2; 4 a(jk) = 4 a(j) 4 a(k) < (jk)^2.
Corollary: a(n) <=  A198442(n).

R. Crandall, The Poisson equation and "natural" Madelung constants, preprint 2012 (see section 2 of BBCZ below).

Jason Kimberley, Table of n, a(n) for n = 3..10000
D. H. Bailey, J. Borwein, R. Crandall and J. Zucker, Lattice sums arising from the Poisson equation, preprint (2012).
D. H. Bailey, J. M. Borwein and J. S. Kimberley, with an appendix by W. B. Ladd, Computer discovery and analysis of large Poisson polynomials, Experimental Mathematics, August 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
OEIS (Plot 2), Plot of (log n, log a(n))
OEIS (Plot 2), Plot of (n, a(n)) and (n, A198442(n))

Cf. A079458, A198442.

A218147 := func; // Jason Kimberley, Oct 23 2012

A218147 := func)/4>; // Jason Kimberley, Nov 14 2015

a(n) = A079458(n) / 4, for n > 2. - Jason Kimberley, Nov 14 2015
Watson Ladd has proved that the sequence satisfies the following recurrence relations, which were conjectured by Jason Kimberley:
a(1) = 1/4, a(2) = 1/2, for notational convenience;
a(4k+1) = (2k)*(2k) for prime 4k+1;
a(4k+3) = (2k+1)*(2k+2) for prime 4k+3;
a(p^2 k) = p^2 * a(p*k) for prime p;
a(jk) = 4*a(j)*a(k) for j coprime to k.

Entry revised by N. J. A. Sloane, May 15 2016, to take into account the fact that the conjectured formula for this sequence has now been established by Watson Ladd.

A302254 Exponent of the group of the Gaussian integers in a reduced system modulo (1+i)^n.

Original entry on oeis.org

1, 1, 2, 4, 4, 4, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152, 4194304, 4194304, 8388608, 8388608

Offset: 0

Jianing Song, Apr 04 2018

For n > 0, the number of elements in the group of the Gaussian integers in a reduced system modulo (1+i)^n is 2^(n-1).

			For Gaussian integer x such that (x, 1+i) = 1, x^4 - 1 = (x + 1)(x - 1)(x + i)(x - i) provides at least 7 factors of 1+i in total (and exactly 7 when x = 2+i), so a(7) = 4.

Cf. A016116, A079458, A227334.

[1,1,2,4,4,4] cat [2^(Floor(n div 2)-1): n in [6..50]]; // Vincenzo Librandi, Apr 04 2018

Join[{1, 1, 2, 4, 4, 4}, Table[2^(Floor[n/2] - 1), {n, 6, 50}]] (* Vincenzo Librandi, Apr 04 2018 *)

For n > 5, a(n) = 2^(floor(n/2) - 1).

For even n, a(n) = A227334(2^(n/2)).

A360323 a(n) is the number of solutions to gcd(a^2 + b^2, p) = 1 where p is the n-th prime and 0 <= a,b <= p-1.

Original entry on oeis.org

2, 8, 16, 48, 120, 144, 256, 360, 528, 784, 960, 1296, 1600, 1848, 2208, 2704, 3480, 3600, 4488, 5040, 5184, 6240, 6888, 7744, 9216, 10000, 10608, 11448, 11664, 12544, 16128, 17160, 18496, 19320, 21904, 22800, 24336, 26568, 27888, 29584, 32040, 32400, 36480

Offset: 1

Paavan Mayurkumar Parekh and Param Mayurkumar Parekh, Feb 03 2023

nonn

The prime numbers can be divided into 3 classes as follows, where 0 <= a,b <= p-1.
p = 2: The solutions are (0,1), (1,0).
p == 1 (mod 4): The number of solutions = p^2 - (number of solutions to a^2 + b^2 == 0 (mod p)). These primes can be written as the sum of two squares, so p = a^2 + b^2 == 0 (mod p). Hence, the number of possible values of (a,b) such that a^2 + b^2 == 0 (mod p) is 2*p - 1, so the final answer is p^2 - (2*p - 1) = (p-1)^2.
p == 3 (mod 4): These primes can't be written as the sum of two squares, so the number of possible values of (a,b) such that a^2 + b^2 == 0 (mod p) is 1 (that is, (0,0) only). Hence, the number of solutions for this case is p^2 - 1.

			a(2) = A079458(A000040(2)) = A079458(3) = 8.

Cf. A000040, A079458.

a(n) = A079458(A000040(n)).

cp's OEIS Frontend

A204617 Multiplicative with a(p^e) = p^(e-1)*H(p). H(2) = 1, H(p) = p - 1 if p == 1 (mod 4) and H(p) = p + 1 if p == 3 (mod 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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A218147 Degree of minimal polynomial satisfied by exp(8*Pi*phi_2(1/n,1/n)), where phi_2 is defined in the Comments.

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A302254 Exponent of the group of the Gaussian integers in a reduced system modulo (1+i)^n.

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Magma

Mathematica

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A360323 a(n) is the number of solutions to gcd(a^2 + b^2, p) = 1 where p is the n-th prime and 0 <= a,b <= p-1.

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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.

A218147 Degree of minimal polynomial satisfied by exp(8Piphi_2(1/n,1/n)), where phi_2 is defined in the Comments.