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.

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A076341 Imaginary 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

0, 0, -1, 0, 1, -2, -1, 0, -8, 2, -1, -4, 1, -2, 0, 0, 1, -16, -1, 4, -12, -2, -1, -8, 8, 2, -47, -4, 1, 0, -1, 0, -16, 2, 4, -32, 1, -2, -8, 8, 1, -24, -1, -4, -17, -2, -1, -16, -16, 16, -12, 4, 1, -94, 8, -8, -24, 2, -1, 0, 1, -2, -79, 0, 16, -32, -1, 4, -28, 8, -1, -64, 1, 2, 17, -4
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			n=21: 21 = 3*7 = (4-1)*(8-1) = (4,-1)*(8,-1) -> (32-(-1)*(-1),-4+(-8)) = (31,-12), therefore a(21)=-12, A076340(21)=31;
n=35: 35 = 5*7 = (4+1)*(8-1) = (4,1)*(8,-1) -> (32-1*(-1),-4+8) = (33,4), therefore a(35)=4, A076340(35)=33.

Crossrefs

Real part = A076340.
Cf. A070750, A076344, A076346, A076348, A076350, A076351.
Cf. A000040, A001358, A000961, A005117, A000290.

Programs

  • Mathematica
    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_] := Im[b[n]];
Array[a, 100] (* Jean-François Alcover, Dec 12 2021 *)

Formula

a(A000040(n)) = A070750(n).
a(A001358(n)) = A076344(n).
a(A000961(n)) = A076346(n).
a(A005117(n)) = A076348(n).
a(A000290(n)) = A076350(n);
a(A076351(n)) = 0.

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.

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.

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.

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Oct 08 2002

Keywords

Examples

			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.

Crossrefs

Cf. A005117, A076340, A076341, A076348.

Programs

  • Mathematica
    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 *)

Extensions

More terms from Amiram Eldar, Feb 24 2024
Showing 1-5 of 5 results.

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