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 1-3 of 3 results.

A076486 Solutions to gcd(sigma(x), phi(x)) < gcd(sigma(core(x)), phi(core(x))), i.e., when A009223(x) < A066086(x) or if A066087(x) < 0.

Original entry on oeis.org

9, 25, 28, 36, 45, 50, 52, 75, 76, 81, 84, 90, 98, 100, 117, 121, 124, 144, 148, 150, 153, 156, 175, 180, 208, 225, 228, 234, 242, 244, 245, 252, 261, 268, 275, 289, 292, 300, 304, 306, 316, 324, 325, 333, 338, 360, 364, 369, 372, 380, 388, 392, 400, 405, 412
Offset: 1

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Author

Labos Elemer, Oct 17 2002

Keywords

Examples

			For n=9: sigma(9)=13, phi(9)=6, gcd(13,6)=1, core(9)=3, sigma(3)=4, phi(3)=2, gcd(4,2)=2.

Links

Crossrefs

Cf. A000010, A000203, A007947, A009223, A023900, A048250, A066086, A066087, A076485.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Do[s1=g1[n]; s2=g2[n]; If[Greater[s2, s1], Print[n]], {n, 1, 256}]

A076487 Solutions to gcd(sigma(x), phi(x)) = gcd(sigma(core(x)), phi(core(x))), i.e., when A009223(x) = A066086(x) or if A066087(x) = 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 77, 78, 79, 80, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95
Offset: 1

Views

Author

Labos Elemer, Oct 17 2002

Keywords

Comments

The squarefree numbers are a subset of this sequence.

Examples

			For n=20: sigma(20)=42, phi(20)=8, gcd(42,8)=2, core(20)=10, sigma(10)=18, phi(10)=4, gcd(18,4)=2, so A009223(20) = A066086(20)=2.

Links

Crossrefs

Cf. A000010, A000203, A007947, A009223, A023900, A048250, A066086, A066087, A076485, A076486.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Do[s1=g1[n]; s2=g2[n]; If[Equal[s2, s1], Print[n]], {n, 1, 256}]
  • PARI
    isok(n) = my(c=core(n)); gcd(sigma(n), eulerphi(n)) == gcd(sigma(c), eulerphi(c)); \\ Michel Marcus, Jul 30 2017

A076488 Nonsquarefree solutions to gcd(sigma(x), phi(x)) = gcd(sigma(core(x)), Phi(core(x))), i.e., when A009223(x) = A066086(x) or if A066087(x)=0 and mu(x)=0.

Original entry on oeis.org

4, 8, 16, 20, 27, 32, 40, 60, 63, 64, 68, 80, 104, 120, 126, 128, 136, 160, 164, 171, 189, 204, 212, 220, 232, 240, 243, 256, 260, 272, 279, 294, 296, 312, 315, 320, 340, 342, 343, 350, 351, 356, 363, 375, 378, 387, 404, 408, 416, 424, 464, 476, 480, 492, 512
Offset: 1

Views

Author

Labos Elemer, Oct 17 2002

Keywords

Examples

			n=60: sigma(60)=168, phi(60)=16, gcd(168,16)=8, core(60)=30, sigma(30)=72, phi(30)=8, gcd(72,8)=8, so A009223(60)=A066086(60)=8.

Links

Crossrefs

Cf. A066086, A066087, A048250, A023900, A000203, A007947, A000010, A009223, A076485-A076487.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Do[s1=g1[n]; s2=g2[n]; If[Equal[s2, s1]&&Equal[MoebiusMu[n], 0], Print[n]], {n, 1, 1024}]
Showing 1-3 of 3 results.

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