A076341 -id:A076341 - cp's OEIS frontend

A076352 Squarefree numbers k such that A076341(k) = 0.

1, 2, 15, 30, 143, 286, 2145, 3599, 4290, 5183, 7198, 10366, 11663, 23326, 32399, 36863, 51983, 53985, 57599, 64798, 73726, 77745, 97343, 103966, 107970, 115198, 121103, 155490, 174945, 176399, 186623, 194686, 242206, 349890, 352798, 359999, 373246, 435599, 485985

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

			Applying the map as defined in A076340, A076341:
A005117(19) = 30 = 5*3*2 = (4+1)*(4-1)*2 -> (4,1)*(4,-1)*(2,0) = (4*4+1,4-4)*(2,0) = (34,0), therefore A076340(30) = 34 and A076341(30) = 0, hence 30 is a term.
A005117(28) = 42 = 7*3*2 = (8-1)*(4-1)*2 -> (8,-1)*(4,-1)*(2,0) = (8*4-1,-8-4)*(2,0) = (62,-24), therefore A076340(42) = 62 and A076341(42) = -24, hence 42 is not a term.

Cf. A005117, A076340, A076341, A076348.

f[p_, e_] := 4*(Floor[p/4] + Floor[Mod[p, 4]/2]) + (2 - Mod[p, 4])*I; f[2, e_] := 2; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[500000], SquareFreeQ[#] && Im[s[#]] == 0 &] (* Amiram Eldar, Feb 24 2024 *)

More terms from Amiram Eldar, Feb 24 2024

A070750 0 if n-th prime is even, 1 if n-th prime is == 1 (mod 4), and -1 if n-th prime is == 3 (mod 4).

Original entry on oeis.org

0, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1

Offset: 1

Reinhard Zumkeller, May 04 2002

Also, sin(prime(n)*Pi/2), where prime(n) = A000040(n), Pi=3.1415... (original definition).
Also imaginary part of primes mapped as defined in A076340, A076341: a(n) = A076341(A000040(n)), real part = A076342.
Legendre symbol (-1/prime(n)) for n > 1. - T. D. Noe, Nov 05 2003
For n > 1, let p = prime(n) and m = (p-1)/2. Then c(m) - a(n) == 0 (mod p), where c(m) = (2*m)!/(m!)^2 = A000984(m) is the central binomial coefficient. [Proof: By definition, c(m)*(m!)^2 - (p-1)! = 0 and therefore c(m)*(m!)^2*(-1)^(m+1) - (p-1)!*(-1)^(m+1) = 0. Now apply Wilson's theorem, (p-1)! == 1 (mod p), and its corollary, (m!)^2 == (-1)^(m+1) (mod p), and finally use the formula by T. D. Noe listed below to replace (-1)^m with a(n).] Similarly, C_m - 2*a(n) == 0 (mod p), with C_m = A000108(m) being the m-th Catalan number. [Proof: By definition, C_m*(p+1)*(m!)^2 - 2*(p-1)! = 0. The result follows proceeding as in the first proof.] - Stanislav Sykora, Aug 11 2014

			p = 4*k+1 (see A002144): a(p) = sin((4*k+1)*Pi/2) = sin(2*k*Pi + Pi/2) = sin(Pi/2) = 1.
p = 4*k+3 (see A002145): a(p) = sin((4*k+3)*Pi/2) = sin(2*k*Pi + 3*Pi/2) = sin(3*Pi/2) = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre Symbol.
Wikipedia, Wilson's theorem.

Cf. A000040, A039702, A070748, A070749, A002144, A002145, A000108, A000984, A134323, A257834, A076340, A076341.

Cf. A060294, A088538.

a070750 = (2 -) . (`mod` 4) . a000040  -- Reinhard Zumkeller, Feb 28 2012

a[n_] := JacobiSymbol[-1, Prime[n]]; a[1] = 0; Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Oct 05 2012, after T. D. Noe *)
Table[Which[EvenQ[p],0,Mod[p,4]==1,1,True,-1],{p,Prime[Range[80]]}] (* Harvey P. Dale, Mar 16 2020 *)

apply(n->2-n%4,primes(100)) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = 2 - prime(n) mod 4 = 2 - A039702(n).
a(n) = (-1)^((prime(n)-1)/2) for n > 1. - T. D. Noe, Nov 05 2003
From Amiram Eldar, Dec 24 2022: (Start)
Product_{n>=1} (1 - a(n)/prime(n)) = 4/Pi (A088538).
Product_{n>=1} (1 + a(n)/prime(n)) = 2/Pi (A060294). (End)

Wording of definition changed by N. J. A. Sloane, Jun 21 2015

A076340 Real part of the function defined multiplicatively on the complex numbers by 2->(2,0) and p->((floor(p/4)+floor((p mod 4)/2))*4,2-(p mod 4)) for odd primes p.

Original entry on oeis.org

1, 2, 4, 4, 4, 8, 8, 8, 15, 8, 12, 16, 12, 16, 17, 16, 16, 30, 20, 16, 31, 24, 24, 32, 15, 24, 52, 32, 28, 34, 32, 32, 47, 32, 33, 60, 36, 40, 49, 32, 40, 62, 44, 48, 68, 48, 48, 64, 63, 30, 65, 48, 52, 104, 49, 64, 79, 56, 60, 68, 60, 64, 112, 64, 47, 94, 68, 64, 95, 66, 72

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

a(n)>0 for n<2187=3^7, a(2187)=-5816, A076341(2187)=-20047.

			n=21: 21 = 3*7 = (4-1)*(8-1) = (4,-1)*(8,-1) -> (32-(-1)*(-1),-4+(-8)) = (31,-12), therefore a(21)=31, A076341(21)=-12;
n=35: 35 = 5*7 = (4+1)*(8-1) = (4,1)*(8,-1) -> (32-1*(-1),-4+8) = (33,4), therefore a(35)=33, A076341(35)=4.

Imaginary part = A076341.
Cf. A076342, A076343, A076345, A076347, A076349.
Cf. A000040, A001358, A000961, A005117, A000290.

b[n_] := If[n == 1, 1, Product[{p, e} = pe; If[p == 2, 2, ((Floor[p/4] + Floor[Mod[p, 4]/2])*4 + (2 - Mod[p, 4]) I)]^e, {pe, FactorInteger[n]}]];
a[n_] := Re[b[n]];
Array[a, 100] (* Jean-François Alcover, Dec 12 2021 *)

a(A000040(n)) = A076342(n).
a(A001358(n)) = A076343(n).
a(A000961(n)) = A076345(n).
a(A005117(n)) = A076347(n).
a(A000290(n)) = A076349(n).

cp's OEIS Frontend

A076352 Squarefree numbers k such that A076341(k) = 0.

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A070750 0 if n-th prime is even, 1 if n-th prime is == 1 (mod 4), and -1 if n-th prime is == 3 (mod 4).

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A076340 Real part of the function defined multiplicatively on the complex numbers by 2->(2,0) and p->((floor(p/4)+floor((p mod 4)/2))*4,2-(p mod 4)) for odd primes p.

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