A069810 -id:A069810 - cp's OEIS frontend

A217301 Numbers n such that gcd(n, phi(n)) = gcd(phi(n), sigma(n)) = gcd(sigma(n), n) = tau(n).

1, 184, 376, 1336, 1912, 2104, 2872, 3064, 3832, 4024, 4792, 5176, 5752, 5944, 6712, 6904, 7096, 7864, 8824, 9784, 10552, 10936, 11512, 11896, 12472, 12664, 12856, 14584, 14776, 16312, 16504, 16696, 17656, 19192, 19384, 19576, 20344, 21304, 21496, 22644

Offset: 1

Jayanta Basu, Mar 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A069810.

Select[Range[20000], GCD[#, EulerPhi[#]] ==  GCD[EulerPhi[#], DivisorSigma[1, #]] == GCD[#, DivisorSigma[1, #]] == DivisorSigma[0, #] &] (* Jean-François Alcover, Mar 18 2013 *)

A306667 Numbers m such that lcm(tau(m), m) = sigma(m) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of the divisors of k (A000005).

Original entry on oeis.org

1, 6, 32760, 51001180160, 54530444405217553992377326508106948362108928, 133821156044600922812153118065015159487725568, 42274041475824304453686528060845522019324411248640, 48949643430560436794021629524876790263031553747866371344635527168

Offset: 1

Jaroslav Krizek, Mar 04 2019

Numbers m such that A009230(m) = A000203(m).

Subsequence of multiply-perfect numbers (A007691).

			6 is a term because lcm(tau(6), 6) = lcm(4, 6) = 12 = sigma(6).

Giovanni Resta, Table of n, a(n) for n = 1..11 (from A007691 data)

Cf. A000005, A000203, A009230.

Cf. A069810 (gcd(k, sigma(k)) = tau(k)).

[n: n in [1..100000] | LCM(NumberOfDivisors(n), n) eq SumOfDivisors(n)]

a(4)-a(8) computed from A007691 data by Giovanni Resta, Mar 05 2019

A224108 Numbers k such that tau(k) divides k, sigma(k) and phi(k).

Original entry on oeis.org

1, 56, 184, 248, 376, 504, 568, 632, 672, 824, 864, 1016, 1208, 1248, 1336, 1528, 1592, 1656, 1784, 1824, 1912, 2016, 2104, 2168, 2232, 2488, 2688, 2872, 2936, 2976, 3064, 3360, 3384, 3448, 3512, 3552, 3704, 3832, 3896, 3968, 4024, 4128, 4284, 4320, 4792, 4856, 5048

Offset: 1

Jayanta Basu, Mar 31 2013

nonn

4 | a(n) for n > 1. Natural density 0. - Charles R Greathouse IV, Mar 31 2013

Zelinsky (2002) called these terms "rare numbers", and noted that if p and q are distinct primes, not equal to 2,3 or 7, then 672*p*q is a term. He proved that for any k > 0 and for sufficiently large m the number of terms not exceeding m is > k*pi(m), where pi(m) = A000720(m). - Amiram Eldar, Feb 20 2021

			56 is in the sequence because 56 has 8 divisors (1, 2, 4, 7, 8, 14, 28, 56), and 8 is a divisor of 56, as well as of sigma(56) = 120 and of phi(56) = 24.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Intersection of A003601, A020491, and A033950.

Cf. A000005, A000010, A000203, A000720, A069810, A217301.

Select[Range[10000], GCD[DivisorSigma[1, #], #, EulerPhi[#], DivisorSigma[0, #]] == DivisorSigma[0, #] &]
Select[Range[5100],AllTrue[{#,DivisorSigma[1,#],EulerPhi[#]}/ DivisorSigma[ 0,#], IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 02 2019 *)

is(n)=my(t=numdiv(n)); n%t==0 && sigma(n)%t==0 && eulerphi(n)%t==0 \\ Charles R Greathouse IV, Mar 31 2013

A306655 Numbers n such that lcm(sigma(n), n) = tau(n) * sigma(n) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 18, 468, 9360, 10880, 79360, 84480, 387072, 777216, 3801600, 7282688, 15037440, 17418240, 27067392, 65544192, 752903424, 1218032640, 4227842304, 4737761280, 6410638080, 11949932544, 19327057920, 26372530800, 37645171200, 224956569600, 243520929792, 876611248128

Offset: 1

Jaroslav Krizek, Mar 03 2019

nonn

Numbers n such that A009242(n) = A000005(n) * A000203(n) = A064840(n).
Also numbers n such that A017666(n) = denominator(sigma(n)/n) = tau(n) = A000005(n).
a(29) > 10^12. - Giovanni Resta, Mar 04 2019

			18 is a term because lcm(sigma(18), 18) = lcm(39, 18) = 234 = tau(18) * sigma(18) = 6 * 39.

Cf. A000005, A000203, A009242, A064840.

Cf. A069810 (gcd(sigma(n), n) = tau(n)).

[n: n in [1..1000000] | LCM(SumOfDivisors(n), n) eq NumberOfDivisors(n)* SumOfDivisors(n)]

Select[Range[1000000], LCM[DivisorSigma[1, #], #] == DivisorSigma[0, #] * DivisorSigma[1, #]&] (* Vaclav Kotesovec, Mar 04 2019 *)

isok(n) = my(sn = sigma(n)); lcm(sn, n) == sn*numdiv(n); \\ Michel Marcus, Mar 04 2019

a(13)-a(16) from Vaclav Kotesovec, Mar 04 2019
a(17) from Daniel Suteu, Mar 04 2019
a(18)-a(28) from Giovanni Resta, Mar 04 2019

cp's OEIS Frontend

A217301 Numbers n such that gcd(n, phi(n)) = gcd(phi(n), sigma(n)) = gcd(sigma(n), n) = tau(n).

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A306667 Numbers m such that lcm(tau(m), m) = sigma(m) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of the divisors of k (A000005).

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A224108 Numbers k such that tau(k) divides k, sigma(k) and phi(k).

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A306655 Numbers n such that lcm(sigma(n), n) = tau(n) * sigma(n) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).

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Author

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Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions