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.

A066831 Numbers k such that sigma(k) divides sigma(phi(k)).

Original entry on oeis.org

1, 13, 71, 87, 89, 181, 203, 305, 319, 362, 667, 899, 1257, 1363, 1421, 1525, 1711, 1798, 1889, 2407, 2501, 2933, 3103, 4609, 4615, 4687, 4843, 5002, 5191, 6583, 7123, 7625, 7627, 9374, 9947, 10063, 10411, 10991, 11107, 12989, 13543, 13891, 14587
Offset: 1

Views

Author

Benoit Cloitre, Jan 19 2002

Keywords

Comments

For odd n, if sigma(phi(n))/sigma(n)=3 then sigma(phi(2*n))/sigma(2*n)=1. - Vladeta Jovovic, Jan 21 2002.
Comments from Vim Wenders, Nov 01 2006: (Start)
This is almost certainly false for even n. For odd n we have phi(n)=phi(2n) and with sigma(2)=3 trivially sigma(phi(n))/sigma(n)=3 <=> sigma(phi(2n))/sigma(2n) = sigma(phi(n))/3.sigma(n)=1.
But suppose n=2m, m odd: again with phi(2m)=phi(m) and sigma(2)=3, sigma(phi(2m)) / sigma(2m)=3 => sigma(phi( m)) /3sigma( m)=3 => sigma(phi( m)) / sigma( m)=9; and with sigma(4)=7 sigma( phi(4m))/ sigma(4m)=1 => sigma(2phi( m))/7sigma( m)=1 => sigma(2phi( m))/ sigma( m)=7. So we get the condition sigma(phi( m)) / sigma( m)=9 <=> sigma(2phi( m))/ sigma( m)=7 which will fail. So if there is a (very) big odd number n in A066831 (numbers n such that sigma(n) divides sigma(phi(n))) with A066831(n) = 9, the conjecture is wrong. I admit I could not yet find such a number, nor do i really know it exists, i.e., A067385(9) exists. (End)

References

  • R. K. Guy, Unsolved Problems in Number Theory, B42.

Links

Crossrefs

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

Programs

  • Mathematica
    For[ n=1, True, n++, If[ Mod[ DivisorSigma[ 1, EulerPhi[ n ] ], DivisorSigma[ 1, n ] ]==0, Print[ n ] ] ]
Select[Range[15000],Divisible[DivisorSigma[1, EulerPhi[#]], DivisorSigma[1,#]]&] (* Harvey P. Dale, Oct 19 2011 *)
  • PARI
    isok(k) = { sigma(eulerphi(k)) % sigma(k) == 0 } \\ Harry J. Smith, Mar 30 2010

Extensions

More terms from Vladeta Jovovic and Robert G. Wilson v, Jan 20 2002
Edited by Dean Hickerson, Jan 20 2002

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

A066881 Numbers n such that sigma(phi(n))/sigma(n) is an integer >= 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, 15331313
Offset: 1

Views

Author

Jason Earls, Jan 22 2002

Keywords

Comments

Conjecture: all natural numbers 1,2,3,...,n will eventually occur among the integer values of sigma(phi(n))/sigma(n).

Examples

			a(1) = 121679 because sigma(phi(121679))/sigma(121679) = 4.

Links

Crossrefs

Cf. A066817.
Cf. A067383, A067382. - R. J. Mathar, Oct 01 2008

Programs

  • Maple
    for n from 1 do spn := numtheory[sigma](numtheory[phi](n)) ; sn := numtheory[sigma](n) ; if spn mod sn = 0 then if spn/sn >= 4 then print(n,spn/sn) fi; fi; od: # R. J. Mathar, Oct 01 2008
  • PARI
    { n=0; for (m=1, 10^10, if((e=sigma(eulerphi(m))) % (s=sigma(m)) == 0 && e/s >= 4, write("b066881.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Apr 03 2010

Extensions

Terms a(16) etc. from R. J. Mathar, Oct 01 2008
Showing 1-6 of 6 results.

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