A076342 -id:A076342 - cp's OEIS frontend

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).

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).

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.

A082542 a(n) = prime(n) + 2 - (prime(n) mod 4).

Original entry on oeis.org

2, 2, 6, 6, 10, 14, 18, 18, 22, 30, 30, 38, 42, 42, 46, 54, 58, 62, 66, 70, 74, 78, 82, 90, 98, 102, 102, 106, 110, 114, 126, 130, 138, 138, 150, 150, 158, 162, 166, 174, 178, 182, 190, 194, 198, 198, 210, 222, 226, 230, 234, 238, 242, 250, 258, 262, 270, 270, 278

Offset: 1

Reinhard Zumkeller, May 02 2003

For k > 1: a(k+1) = a(k) if and only if prime(k) == 1 modulo 4 and prime(k+1) = prime(k) + 2, see A071695 and A071696.

			a(2) = 2 because the second prime is 3, and 3 + 2 - 3 = 2.
a(3) = 6 because the third prime is 5, and 5 + 2 - 1 = 6.
a(4) = 6 because the fourth prime is 7, and 7 + 2 - 3 = 6.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A060294, A070750, A076342.

Cf. A039702, A071695, A071696, A100484.

[2 + NthPrime(n) - (NthPrime(n) mod 4): n in [1..60]]; // G. C. Greubel, Nov 14 2018

Table[Prime[n] + 2 - Mod[Prime[n], 4], {n, 60}] (* Alonso del Arte, Feb 23 2015 *)
#+2-Mod[#,4]&/@Prime[Range[60]] (* Harvey P. Dale, Aug 24 2025 *)

vector(60, n, 2 + prime(n) - lift(Mod(prime(n),4))) \\ G. C. Greubel, Nov 14 2018

a(n) = A000040(n) + A070750(n).
a(n+1) = p + (-1/p) = p + (-1)^((p-1)/2), where p is the n-th odd prime and (-1/p) denotes the value of Legendre symbol. - Lekraj Beedassy, Mar 17 2005
a(n) = (A000040(n) OR 3) - 1. - Jon Maiga, Nov 14 2018
From Amiram Eldar, Dec 24 2022: (Start)
a(n) = A100484(n) - A076342(n).
Product_{n>=1} a(n)/prime(n) = 2/Pi (A060294). (End)

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.

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.

A261208 Terms of the Leibniz formula (as Euler product) that generate successively better approximations to Pi.

Original entry on oeis.org

1, 3, 4, 5, 8, 47, 49, 95, 247, 251, 253, 742, 4268, 4270, 4288, 11445, 30123, 30701, 30703, 62592, 62690, 62992, 3535871, 3535872, 3664203, 3664204, 3664214, 3664220, 3665670, 3665696, 3665842, 3665854, 3665866, 3708907, 3708909, 3708913, 3708929, 3708931, 3708935, 3708957, 3708983, 3708985, 3709017

Offset: 1

Steven Lubars, Aug 11 2015

nonn

			Calculating the first 8 terms: c(1)=3, c(2)=3.75, c(3)=3.28125, c(4)=3.0078125, c(5)=3.2584635416, c(6)=3.462117513020833, c(7)=3.289011637369791, c(8)=3.1519694858127165.
In the above sequence, terms 1, 3, 4, 5, and 8 provide successively closer approximations of Pi (whereas approximations 2, 6, and 7 do not).

Steven Lubars, Table of n, a(n) for n = 1..71
Wikipedia, Euler Product

Cf. A065091, A076342.

s Pi=3.141592653589793238,a=3,n=1,d=Pi-a
w !,1
f i=6:6:1e10 d
  s L=i+1**.5\1
  f j=i-1:2:i+1 d
    f k=3:2:L q:'(j#k)
    i j#k d
      s a=a*j/(j#4+j-2),n=n+1
      i $FN(Pi-a,"-")Steven Lubars, Aug 14 2015

nearmul(p) = if (p % 4 == 1, p-1, p+1);
lista(nn) = {print1(lb = 1, ", "); v = 3; ld = abs(Pi-3); for (n=2, nn, np = prime(n+1); v *= np/nearmul(np); if ((nld=abs(Pi-v)) < ld, print1(n, ", "); ld = nld););} \\ Michel Marcus, Aug 14 2015

Pi = 4*b(1)*b(2)*b(3)*... where b(n) is the n-th odd prime (A065091) divided by its nearest multiple of 4.

Let c(n) be the n-th term of the expansion such that c(n) = 4*b(1)*...*b(n). The sequence consists of the values n such that c(n) provides a closer approximation of Pi than previous approximations c(1)...c(n-1).

cp's OEIS Frontend

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

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A082542 a(n) = prime(n) + 2 - (prime(n) mod 4).

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

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

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A261208 Terms of the Leibniz formula (as Euler product) that generate successively better approximations to Pi.

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MUMPS

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Formula