A067385 -id:A067385 - cp's OEIS frontend

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

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

Benoit Cloitre, Jan 19 2002

nonn

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)

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

Harry J. Smith, Table of n, a(n) for n = 1..1000

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

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 *)

isok(k) = { sigma(eulerphi(k)) % sigma(k) == 0 } \\ Harry J. Smith, Mar 30 2010

More terms from Vladeta Jovovic and Robert G. Wilson v, Jan 20 2002

Edited by Dean Hickerson, Jan 20 2002

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

Dean Hickerson, Jan 20 2002

nonn

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

Donovan Johnson, Table of n, a(n) for n = 1..1000

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

For[ n=1, True, n++, If[ DivisorSigma[ 1, EulerPhi[ n ] ]/DivisorSigma[ 1, n ]==2, Print[ n ] ] ]

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

Dean Hickerson, Jan 20 2002

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

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

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 *)

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

Dean Hickerson, Jan 20 2002

nonn

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

Donovan Johnson, Table of n, a(n) for n = 1..500

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

For[ n=1, True, n++, If[ DivisorSigma[ 1, EulerPhi[ n ] ]/DivisorSigma[ 1, n ]==4, Print[ n ] ] ]

is(n)=sigma(phi(n=factor(n)))/sigma(n)==4 \\ Charles R Greathouse IV, Nov 27 2013

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

Donovan Johnson, Oct 19 2011

nonn

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

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

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

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

A067740 Smallest number k such that sigma(k)/sigma(phi(k)) = n.

Original entry on oeis.org

1, 14, 2, 6, 118740, 2915640, 74322349920

Offset: 1

Labos Elemer, Jan 29 2002

The quotient sigma(k)/sigma(phi(k)) is integral for the numbers in A190503. Does a(n) exist for all n?

10^11 < a(8) <= 11224976029787520. - Donovan Johnson, Jun 07 2011

			n=6, a(6)=2915640, sigma(2915640)=11793600, phi(2915640)=608256, sigma(608256)=1965600 and 11793600=6*1965600.

Cf. A000203, A000010, A067385, A190503.

g[x_] := DivisorSigma[1, x] / DivisorSigma[1, EulerPhi[x]]; m=10; up=200000; a = Table[0, {m}]; Do[ b = g[n]; If[b <= m && IntegerQ[b] && a[[b]] == 0, a[[b]] = n], {n, 1, up} ]; a

a(n)=Min{x; A000203(x)/A000203[A000010(x)]=n}

a(7) from Donovan Johnson, Jun 07 2011

cp's OEIS Frontend

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

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

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

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

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

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A067740 Smallest number k such that sigma(k)/sigma(phi(k)) = n.

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