A033631 -id:A033631 - cp's OEIS frontend

A115619 Sigma(A033631(n)) {sigma is the sum of divisors function A000203}.

1, 120, 546, 1680, 2880, 7812, 14520, 34200, 40320, 40320, 53760, 68796, 63360, 76608, 78624, 80640, 89280, 86400, 95040, 90720, 129600, 178560, 155496, 205920, 214272, 190080, 223200, 245520, 241920, 244800, 267840, 280800, 259200, 312480, 345600, 313560, 393120, 374976, 401760, 367200, 464256

Offset: 1

Lekraj Beedassy, Jan 26 2006

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

for(n=1,1e6,if(sigma(eulerphi(n))==sigma(n),print1(sigma(n)", "))) \\ Charles R Greathouse IV, Feb 19 2013

A115620 Phi(A033631(n)) {phi is the Euler totient function A000010}.

Original entry on oeis.org

1, 56, 180, 836, 840, 2400, 4536, 10584, 10920, 12540, 17388, 18720, 23736, 24864, 25500, 26220, 23760, 32376, 39560, 45356, 53960, 49680, 77744, 71208, 60720, 87032, 62640, 69120, 63840, 91776, 76560, 97128, 118712, 88560, 83160

Offset: 1

Lekraj Beedassy, Jan 26 2006

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Corrected and extended by Franklin T. Adams-Watters, May 12 2006

A108510 Primes p such that 3^A000027(7p) is a subsequence of A033631.

Original entry on oeis.org

1979, 2699, 7649, 1131569, 2482199, 2886839, 90425537, 2774476799

Offset: 1

Farideh Firoozbakht, Jun 07 2005

There is no further term up to prime(300000000).

			Let n be a natural number and m=3^n*7*90425537;
then sigma(phi(m)) = sigma(3^(n-1)*2*6*90425536) = sigma(3^n*2^8*677*2087) = sigma(3^n)*8*90425538 = sigma(3^n)*sigma(7)*sigma(90425537) = sigma(3^n*7*90425537) = sigma(m),
hence m is in A033631 and the prime number 90425537 is a term of this sequence.

Cf. A000027, A033631.

A006872 Numbers k such that phi(k) = phi(sigma(k)).

Original entry on oeis.org

1, 3, 15, 26, 39, 45, 74, 104, 111, 117, 122, 146, 175, 183, 195, 219, 296, 314, 333, 357, 386, 471, 488, 549, 554, 555, 579, 584, 585, 608, 626, 646, 657, 794, 831, 842, 914, 915, 939, 962, 1071, 1082, 1095, 1191, 1226, 1256, 1263, 1292, 1322, 1346

Offset: 1

Jeffrey Shallit, N. J. A. Sloane

nonn

S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
R. K. Guy, Unsolved Problems in Number Theory, B42.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe, terms 1001..12394 from Marius A. Burtea)
S. W. Golomb, Letter to N. J. A. Sloane, Oct. 1992
S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)

Cf. A000010, A000203, A062401, A353637 (characteristic function).
Positions of zeros in A353636.
Setwise difference of A353684 and A353683, and also of A353685 and A353686.
Intersection of A353684 and A353685.
Subsequences: A260021, A353634, A353635, A353679 (odd terms).
Cf. also A033631, A033632, A067880, A137815, A260020.

a006872 n = a006872_list !! (n-1)
a006872_list = filter (\x -> a000010' x == a000010' (a000203' x)) [1..]
-- Reinhard Zumkeller, Jul 14 2015

[n:n in [1..2000]| EulerPhi(SumOfDivisors(n)) eq EulerPhi(n)]; // Marius A. Burtea, Jan 01 2019

Select[Range@ 1350, EulerPhi@ # == EulerPhi@ DivisorSigma[1, #] &] (* Michael De Vlieger, Jan 01 2019 *)

lista(nn) = {for (i=1, nn, if (eulerphi(i)==eulerphi(sigma(i)), print1(i, ", ")););} \\ Michel Marcus, May 25 2013

More terms from Jud McCranie

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

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

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

Dean Hickerson, Jan 20 2002

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

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

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

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

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

cp's OEIS Frontend

A115619 Sigma(A033631(n)) {sigma is the sum of divisors function A000203}.

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A115620 Phi(A033631(n)) {phi is the Euler totient function A000010}.

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A108510 Primes p such that 3^A000027*(7*p) is a subsequence of A033631.

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A006872 Numbers k such that phi(k) = phi(sigma(k)).

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

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A108510 Primes p such that 3^A000027(7p) is a subsequence of A033631.