cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			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.

Crossrefs

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			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.

Crossrefs

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			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.

Crossrefs

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			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.

Crossrefs

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Crossrefs

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Crossrefs

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			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.

Crossrefs

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			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.

Crossrefs

Real part = A076345, A070750.

A076351 Numbers n such that A076341(n)=0.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 30, 32, 60, 64, 120, 128, 143, 225, 240, 256, 286, 450, 480, 512, 572, 900, 960, 1024, 1144, 1800, 1920, 2048, 2145, 2288, 3375, 3599, 3600, 3840, 4096, 4290, 4576, 5183, 6750, 7198, 7200, 7680, 8192, 8580, 9152
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			Applying the map as defined in A076340, A076341:
120=5*3*2^3=(4+1)*(4-1)*2^3 -> (4,1)*(4,-1)*(2,0)^3 = (4*4+1,4-4)*(8,0) = (136,0), therefore A076340(120)=136 and A076341(120)=0;
130=13*5*2=(12+1)*(4+1)*2 -> (12,1)*(4,1)*(2,0) = (12*4-1,12+4)*(2,0) = (94,32), therefore A076340(130)=94 and A076341(130)=32, hence 130 is not a term.
Showing 1-10 of 13 results. Next

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