A066930 -id:A066930 - cp's OEIS frontend

A073858 Numbers k such that sigma(phi(k)) divides phi(sigma(k)).

1, 2, 4, 9, 16, 18, 64, 100, 225, 242, 450, 516, 729, 1458, 3872, 4096, 4624, 13932, 14406, 17672, 18225, 20124, 21780, 28900, 29262, 29616, 36450, 45996, 62500, 65025, 65536, 76832, 92778, 95916, 106092, 106308, 114630, 114930

Offset: 1

Benoit Cloitre, Sep 02 2002

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

Cf. A033632, A066930.

Select[Range[115000],Divisible[EulerPhi[DivisorSigma[1,#]],DivisorSigma[ 1,EulerPhi[ #]]]&] (* Harvey P. Dale, Jan 31 2021 *)

isok(k) = eulerphi(sigma(k)) % sigma(eulerphi(k))==0 \\ Donovan Johnson, Jul 05 2012

A370689 Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

Original entry on oeis.org

1, 1, 3, 1, 7, 3, 3, 7, 1, 7, 9, 7, 14, 3, 15, 1, 31, 1, 39, 5, 7, 3, 9, 15, 7, 7, 39, 7, 7, 5, 9, 31, 21, 31, 15, 7, 91, 39, 5, 31, 15, 7, 24, 7, 5, 3, 9, 31, 8, 7, 21, 10, 49, 39, 15, 15, 91, 7, 45, 31, 28, 9, 91, 1, 31, 7, 9, 7, 21, 5, 6, 5, 65, 91, 3, 91, 21

Offset: 1

Amiram Eldar, Feb 27 2024

			Fractions begin with: 1, 1/2, 3/2, 1/2, 7/2, 3/4, 3, 7/8, 1, 7/6, 9/2, 7/12, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.

Cf. A000010, A000203, A033632, A062401, A062402, A065395, A066930, A289336, A073858 (positions of 1's), A289412, A370690 (denominators).

Cf. A080130, A091724.

Table[DivisorSigma[1, EulerPhi[n]]/EulerPhi[DivisorSigma[1, n]], {n, 1, 100}] // Numerator

a(n) = {my(f = factor(n)); numerator(sigma(eulerphi(f)) / eulerphi(sigma(f)));}

Let f(n) = a(n)/A370690(n) = A062402(n)/A062401(n).
Formulas from De Koninck and Luca (2007):
lim sup_{n->oo} f(n)/log_2(n)^2 = exp(2*gamma) (A091724).
lim inf_{n->oo} f(n)/log_2(n)^2 = delta exists, and exp(-gamma)/40 <= delta <= 2*exp(-gamma).
Sum_{k=1..n} f(k) = c * exp(2*gamma) * log_3(n)^2 * n + O(n * log_3(n)^(3/2)), where c = Product_{p prime} (1 - 3/(p*(p + 1)) + 1/(p^2*(p + 1)) + ((p-1)^3/p^2)*Sum_{k>=3} 1/(p^k-1)) = 0.45782563109026414241... .

A370690 Denominator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 1, 8, 1, 6, 2, 12, 3, 2, 8, 2, 6, 2, 8, 4, 4, 2, 2, 16, 5, 3, 16, 6, 1, 8, 2, 36, 8, 18, 4, 18, 18, 16, 2, 24, 2, 8, 5, 4, 2, 2, 2, 60, 3, 10, 8, 7, 9, 32, 4, 8, 32, 3, 8, 48, 5, 4, 48, 2, 6, 8, 2, 4, 8, 4, 1, 8, 12, 36, 2, 48, 4, 4, 4, 20, 11

Offset: 1

Amiram Eldar, Feb 27 2024

See A370689 for details.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.

Cf. A000010, A000203, A033632, A062401, A062402, A065395, A066930 (positions of 1's), A073858, A289336, A289412, A370689 (numerators).

Table[DivisorSigma[1, EulerPhi[n]]/EulerPhi[DivisorSigma[1, n]], {n, 1, 100}] // Denominator

a(n) = {my(f = factor(n)); denominator(sigma(eulerphi(f)) / eulerphi(sigma(f)));}

cp's OEIS Frontend

A073858 Numbers k such that sigma(phi(k)) divides phi(sigma(k)).

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A370689 Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

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A370690 Denominator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

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Mathematica

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