A076340 -id:A076340 - cp's OEIS frontend

A076342 a(n) = A076340(A000040(n)), real part of primes mapped as defined in A076340, A076341.

2, 4, 4, 8, 12, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 60, 60, 68, 72, 72, 80, 84, 88, 96, 100, 104, 108, 108, 112, 128, 132, 136, 140, 148, 152, 156, 164, 168, 172, 180, 180, 192, 192, 196, 200, 212, 224, 228, 228, 232, 240, 240, 252, 256, 264, 268, 272

Offset: 1

Reinhard Zumkeller, Oct 08 2002

By definition of the map defined in A076340, A076341: 2->(2,0) and p->((floor(p/4)+floor((p mod 4)/2))*4,2-(p mod 4)) for odd primes p.

Number of solutions to x^2 + y^2 = 1 (mod p). - Lekraj Beedassy, Oct 22 2004

			A000040(11)=31=(32-1) -> (32,-1), therefore a(11)=32 and A070750(11)=-1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A070750, A076340, A076341, A082542, A088538.

f:= proc(n) local p;
  p:= ithprime(n);
  if p mod 4 = 1 then p-1 elif p mod 4 = 3 then p+1 else 2 fi
end proc:
map(f, [$1..100]); # Robert Israel, Dec 26 2016

a[1] = 2; a[n_] := With[{p = Prime[n]}, p - JacobiSymbol[-1, p]]; Array[a, 60] (* Jean-François Alcover, Feb 01 2018, after Lekraj Beedassy *)
a[n_] := Prime[n] - 2 + Mod[Prime[n], 4]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

a(n) = p-(-1/p) = p+(-1)^{(p+1)/2} for an odd prime p. {(a/b) stands for the value of the Legendre symbol}. - Lekraj Beedassy, Oct 22 2004
From Amiram Eldar, Dec 24 2022: (Start)
a(n) = A000040(n) - A070750(n).
a(n) = A100484(n) - A082542(n).
Product_{n>=1} a(n)/prime(n) = 4/Pi (A088538). (End)

A076348 A076341(A005117(n)), imaginary part of squarefree numbers mapped as defined in A076340, A076341.

Original entry on oeis.org

0, 0, -1, 1, -2, -1, 2, -1, 1, -2, 0, 1, -1, -12, -2, -1, 2, 1, 0, -1, -16, 2, 4, 1, -2, -8, 1, -24, -1, -2, -1, -12, 1, 8, -24, 2, -1, 1, -2, 16, -32, -1, -28, 8, -1, 1, 2, -20, -16, -1, 2, -1, 20, -2, -24, 1, -4, -36, -2, 16, 1, 1, -24, -1, -17, 2, -1, 1, 16, -32, 1, -48, 20, -2, -8

Offset: 1

Reinhard Zumkeller, Oct 08 2002

sign

			Applying the map as defined in A076340, A076341:
A005117(40)=65=5*13=(4+1)*(12+1) -> (4,1)*(12,1) = (4*12-1,12+4) = (47,16), therefore a(40)=16 and A076347(40)=47.

Real part = A076347, A070750, A076344, A076350, A076352.

A076347 A076340(A005117(n)), real part of squarefree numbers mapped as defined in A076340, A076341.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 8, 12, 12, 16, 17, 16, 20, 31, 24, 24, 24, 28, 34, 32, 47, 32, 33, 36, 40, 49, 40, 62, 44, 48, 48, 65, 52, 49, 79, 56, 60, 60, 64, 47, 94, 68, 95, 66, 72, 72, 72, 95, 98, 80, 80, 84, 63, 88, 113, 88, 97, 127, 96, 81, 96, 100, 130, 104, 136, 104, 108, 108

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

			Applying the map as defined in A076340, A076341:
A005117(40)=65=5*13=(4+1)*(12+1) -> (4,1)*(12,1) = (4*12-1,12+4) = (47,16), therefore a(40)=47 and A076348(40)=16.

Imaginary part = A076348, A076342, A076343, A076349.

A076343 A076340(A001358(n)), real part of semiprimes mapped as defined in A076340, A076341.

Original entry on oeis.org

4, 8, 15, 8, 16, 17, 31, 24, 15, 24, 47, 32, 33, 40, 49, 48, 63, 65, 49, 79, 56, 64, 47, 95, 72, 95, 80, 63, 88, 113, 97, 127, 96, 81, 104, 145, 97, 120, 129, 143, 120, 161, 175, 159, 136, 191, 144, 145, 111, 144, 129, 160, 209, 191, 168, 143, 239, 176, 241, 143

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

			Applying the map as defined in A076340, A076341:
A001358(15)=39=3*13=(4-1)*(12+1) -> (4,-1)*(12,1) = (4*12+1,4-12) = (49,-8), therefore a(15)=49 and A076344(120)=-8.

Imaginary part = A076344, A076342, A076347.

A076344 A076341(A001358(n)), imaginary part of semiprimes mapped as defined in A076340, A076341.

Original entry on oeis.org

0, -2, -8, 2, -2, 0, -12, -2, 8, 2, -16, 2, 4, -2, -8, -2, -16, -12, 8, -24, 2, -2, 16, -28, 2, -20, 2, 20, -2, -24, -4, -36, -2, 16, 2, -32, 20, -2, -8, -24, 2, -36, -48, -28, -2, -52, -2, 0, 32, 2, 28, -2, -48, -32, -2, 24, -64, 2, -56, 40, -4, 2, -72, 2, -20, 44, -2, -32, -76, -2

Offset: 1

Reinhard Zumkeller, Oct 08 2002

sign

			Applying the map as defined in A076340, A076341:
A001358(15)=39=3*13=(4-1)*(12+1) -> (4,-1)*(12,1) = (4*12+1,4-12) = (49,-8), therefore a(15)=-8 and A076343(15)=49.

Real part = A076343, A070750, A076348.

A076349 A076340(A000290(n)), real part of squares mapped as defined in A076340, A076341.

Original entry on oeis.org

1, 4, 15, 16, 15, 60, 63, 64, 161, 60, 143, 240, 143, 252, 289, 256, 255, 644, 399, 240, 817, 572, 575, 960, 161, 572, 495, 1008, 783, 1156, 1023, 1024, 1953, 1020, 1073, 2576, 1295, 1596, 2337, 960, 1599, 3268, 1935, 2288, 4335, 2300, 2303, 3840, 3713

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

Imaginary part = A076350, A076347.

A076350 A076341(A000290(n)), imaginary part of squares mapped as defined in A076340, A076341.

Original entry on oeis.org

0, 0, -8, 0, 8, -32, -16, 0, -240, 32, -24, -128, 24, -64, 0, 0, 32, -960, -40, 128, -744, -96, -48, -512, 240, 96, -4888, -256, 56, 0, -64, 0, -1504, 128, 264, -3840, 72, -160, -784, 512, 80, -2976, -88, -384, -2312, -192, -96, -2048, -2016, 960, -1560, 384, 104

Offset: 1

Reinhard Zumkeller, Oct 08 2002

sign

Real part = A076349, A076348.

A076345 A076340(A000961(n)), real part of prime powers mapped as defined in A076340, A076341.

Original entry on oeis.org

1, 2, 4, 4, 4, 8, 8, 15, 12, 12, 16, 16, 20, 24, 15, 52, 28, 32, 32, 36, 40, 44, 48, 63, 52, 60, 60, 64, 68, 72, 72, 80, 161, 84, 88, 96, 100, 104, 108, 108, 112, 143, 52, 128, 128, 132, 136, 140, 148, 152, 156, 164, 168, 143, 172, 180, 180, 192, 192, 196, 200, 212

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

			Applying the map as defined in A076340, A076341:
A000961(16)=27=3*3*3=(4-1)*(4-1)*(4-1) -> (4,-1)*(4,-1)*(4,-1) = (4*4-1,-8)*(4,-1) = (4*15-8,-32-15) = (52,-47), therefore a(16)=52 and A076346(16)=-47.

Imaginary part = A076346, A076342.

A076346 A076341(A000961(n)), imaginary part of prime powers mapped as defined in A076340, A076341.

Original entry on oeis.org

0, 0, -1, 0, 1, -1, 0, -8, -1, 1, 0, 1, -1, -1, 8, -47, 1, -1, 0, 1, 1, -1, -1, -16, 1, -1, 1, 0, -1, -1, 1, -1, -240, -1, 1, 1, 1, -1, -1, 1, 1, -24, 47, -1, 0, -1, 1, -1, 1, -1, 1, -1, -1, 24, 1, -1, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1121, -1, 0, 1, -1, 1, -1, 1, 1, -1, 32, 1, -1

Offset: 1

Reinhard Zumkeller, Oct 08 2002

sign

			Applying the map as defined in A076340, A076341:
A000961(16)=27=3*3*3=(4-1)*(4-1)*(4-1) -> (4,-1)*(4,-1)*(4,-1) = (4*4-1,-8)*(4,-1) = (4*15-8,-32-15) = (52,-47), therefore a(16)=-47 and A076345(16)=52.

Real part = A076345, A070750.

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

cp's OEIS Frontend

A076342 a(n) = A076340(A000040(n)), real part of primes mapped as defined in A076340, A076341.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A076348 A076341(A005117(n)), imaginary part of squarefree numbers mapped as defined in A076340, A076341.

Views

Author

Keywords

Examples

Crossrefs

A076347 A076340(A005117(n)), real part of squarefree numbers mapped as defined in A076340, A076341.

Views

Author

Keywords

Examples

Crossrefs

A076343 A076340(A001358(n)), real part of semiprimes mapped as defined in A076340, A076341.

Views

Author

Keywords

Examples

Crossrefs

A076344 A076341(A001358(n)), imaginary part of semiprimes mapped as defined in A076340, A076341.

Views

Author

Keywords

Examples

Crossrefs

A076349 A076340(A000290(n)), real part of squares mapped as defined in A076340, A076341.

Views

Author

Keywords

Crossrefs

A076350 A076341(A000290(n)), imaginary part of squares mapped as defined in A076340, A076341.

Views

Author

Keywords

Crossrefs

A076345 A076340(A000961(n)), real part of prime powers mapped as defined in A076340, A076341.

Views

Author

Keywords

Examples

Crossrefs

A076346 A076341(A000961(n)), imaginary part of prime powers mapped as defined in A076340, A076341.

Views

Author

Keywords

Examples

Crossrefs

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions