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 1-6 of 6 results.

A067382 Numbers n such that sigma(phi(n))/sigma(n) = 2.

Original entry on oeis.org

13, 71, 89, 203, 305, 319, 667, 1363, 1421, 1525, 1711, 1889, 2407, 2933, 3103, 4609, 4615, 4843, 5191, 6583, 7123, 7625, 7627, 9947, 10063, 10411, 11107, 13543, 13891, 14587, 16327, 17023, 19693, 20851, 23075, 24331, 24721, 25027, 25723
Offset: 1

Views

Author

Dean Hickerson, Jan 20 2002

Keywords

Comments

Sequence is infinite. Contains subsequences like 5^i*61, 5^i*13*71, 7^i*29 ... (see also Farideh Firoozbakht's comment on A033631.) - vim(AT)gmx.li, Nov 03 2006

Links

Crossrefs

Cf. A033631, A066831, A067383, A067384, A067385, A197952.

Programs

  • Mathematica
    For[ n=1, True, n++, If[ DivisorSigma[ 1, EulerPhi[ n ] ]/DivisorSigma[ 1, n ]==2, Print[ n ] ] ]
  • PARI
    is(n)=sigma(eulerphi(n=factor(n)))/sigma(n)==2 \\ Charles R Greathouse IV, Nov 27 2013

A067383 Numbers n such that sigma(phi(n))/sigma(n) = 3.

Original entry on oeis.org

181, 899, 2501, 4687, 10991, 12989, 17653, 25199, 25853, 26549, 26657, 54473, 65941, 68381, 72007, 82777, 96197, 98903, 102719, 116449, 124013, 135907, 150121, 169153, 188917, 193553, 201173, 207461, 219559, 234301, 237961, 239279
Offset: 1

Views

Author

Dean Hickerson, Jan 20 2002

Keywords

Links

Crossrefs

Cf. A033631, A066831, A067382, A067384, A067385, A197952.

Programs

  • Mathematica
    For[ n=1, True, n++, If[ DivisorSigma[ 1, EulerPhi[ n ] ]/DivisorSigma[ 1, n ]==3, Print[ n ] ] ]
Select[Range[250000],DivisorSigma[1,EulerPhi[#]]/DivisorSigma[1,#]==3&] (* Harvey P. Dale, Sep 08 2024 *)
  • PARI
    is(n)=sigma(eulerphi(n=factor(n)))/sigma(n)==3 \\ Charles R Greathouse IV, Nov 27 2013

A067384 Numbers n such that sigma(phi(n))/sigma(n) = 4.

Original entry on oeis.org

121679, 1043909, 2350171, 2918263, 3396103, 3566807, 3688067, 4019467, 4562827, 5963407, 7300697, 7485979, 7853933, 8103301, 8364151, 9237779, 9514213, 9638527, 10531123, 11094619, 11384447, 12721937, 13576267
Offset: 1

Views

Author

Dean Hickerson, Jan 20 2002

Keywords

Comments

Subsequence of A066881. - R. J. Mathar, Sep 30 2008

Links

Crossrefs

Cf. A033631, A066831, A067382, A067383, A067385, A197952.

Programs

  • Mathematica
    For[ n=1, True, n++, If[ DivisorSigma[ 1, EulerPhi[ n ] ]/DivisorSigma[ 1, n ]==4, Print[ n ] ] ]
  • PARI
    is(n)=sigma(phi(n=factor(n)))/sigma(n)==4 \\ Charles R Greathouse IV, Nov 27 2013

A067385 a(n) is smallest x such that sigma(phi(x))/sigma(x) = n.

Original entry on oeis.org

1, 13, 181, 121679, 1033474069
Offset: 1

Views

Author

Dean Hickerson, Jan 20 2002

Keywords

Comments

A023199(6) < a(6) <= 1794819234390989. - Donovan Johnson, Oct 24 2011

Crossrefs

Cf. A023199, A033631, A066831, A067382, A067383, A067384, A197952.

Programs

  • Mathematica
    a[ n_ ] := For[ x=1, True, x++, If[ DivisorSigma[ 1, EulerPhi[ x ] ]/DivisorSigma[ 1, x ]==n, Return[ x ] ] ]

Extensions

a(5) from Vim Wenders, Mar 11 2007

A197952 Numbers n such that sigma(phi(n))/sigma(n) = 5.

Original entry on oeis.org

1033474069, 1604277377, 2741806637, 9941342981, 14754456491, 14859359791, 15887724883, 16990353761, 17266051069, 20892536447, 21776951239, 24435763193, 25165559143, 32325726313, 38313868379, 38580669727, 38856433193, 47906215417, 49094416289, 56237053007
Offset: 1

Views

Author

Donovan Johnson, Oct 19 2011

Keywords

Examples

			sigma(phi(25165559143))/sigma(25165559143) = 127671828480/25534365696 = 5.

Crossrefs

Cf. A033631, A066831, A067382, A067383, A067384, A067385.

Programs

  • PARI
    for(n=1033474069, 3*10^10, if(sigma(eulerphi(n))/sigma(n)==5, print1(n, ", ")))

Extensions

a(14)-a(20) from Donovan Johnson, Nov 11 2011

A190503 Numbers k such that sigma(phi(k)) divides sigma(k).

Original entry on oeis.org

1, 2, 6, 12, 14, 22, 24, 28, 44, 46, 48, 56, 68, 87, 88, 92, 94, 96, 112, 118, 166, 174, 176, 184, 188, 192, 214, 224, 236, 332, 334, 352, 358, 362, 368, 376, 384, 390, 410, 428, 448, 454, 472, 526, 664, 668, 694, 704, 716, 718, 736, 752, 766, 768, 856, 896
Offset: 1

Views

Author

T. D. Noe, May 11 2011

Keywords

Comments

These numbers appear indirectly in A067740, which seeks the least k such that sigma(k)/sigma(phi(k)) = n. Most of these numbers are even. The odd terms (1, 87, 1257, 41559, 56679, ...) all appear to produce sigma(k)/sigma(phi(k)) = 1.

Links

Crossrefs

Cf. A000010 (phi), A000203 (sigma), A062402, A067740.
Cf. A033631 (k such that sigma(k)/sigma(phi(k)) = 1).
Cf. A066831 (k such that sigma(k) divides sigma(phi(k))).

Programs

  • Mathematica
    Select[Range[1000], IntegerQ[DivisorSigma[1,#]/DivisorSigma[1,EulerPhi[#]]] &]
  • PARI
    is(k) = {my(f = factor(k), s = sigma(f), p = eulerphi(f)); !(s % sigma(p));} \\ Amiram Eldar, May 17 2024
Showing 1-6 of 6 results.

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