A289336 -id:A289336 - cp's OEIS frontend

A289412 a(n) = denominator of (sigma(n) / phi(n)).

1, 1, 1, 2, 2, 1, 3, 4, 6, 2, 5, 1, 6, 1, 1, 8, 8, 2, 9, 4, 3, 5, 11, 2, 20, 2, 9, 3, 14, 1, 15, 16, 5, 8, 1, 12, 18, 3, 3, 8, 20, 1, 21, 5, 4, 11, 23, 4, 14, 20, 4, 12, 26, 3, 5, 1, 9, 14, 29, 2, 30, 5, 9, 32, 4, 5, 33, 16, 11, 1, 35, 8, 36, 6, 10, 9, 5, 1

Offset: 1

Jaroslav Krizek, Aug 19 2017

a(n) = 1 for numbers in A020492 (balanced numbers).

			For n = 7, sigma(7) / phi(7) = 8/6 = 4/3, a(n) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A289336 (numerators).

Cf. A000010, A000203, A018894, A020492, A023897, A055234, A070032, A070033.

[Denominator(SumOfDivisors(n) / EulerPhi(n)): n in[1..1000]]

Array[Denominator[DivisorSigma[1, #]/EulerPhi[#]] &, 78] (* Michael De Vlieger, Aug 19 2017 *)

a(n) = denominator(sigma(n)/eulerphi(n)); \\ Michel Marcus, Aug 21 2017

a(n) = denominator of (A000203(n) / A000010(n)).

A293391 Integers n such that sigma(n)/phi(n) is a perfect square.

Original entry on oeis.org

1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, 3135, 3596, 3828, 3956, 4064, 5396, 6678, 8636, 10098, 12648, 20026, 20790, 21318, 22152, 23374, 24882, 25714, 26040, 35074, 35343, 39105, 41656, 43890, 44660, 49938, 55154, 56134, 56536, 61344, 71145, 74613, 86304, 87087, 94944

Offset: 1

Jose Arnaldo Bebita Dris, Oct 08 2017

nonn

From Robert Israel, Dec 12 2017: (Start)
Intersection of A011257 and A020492.
If x and y are coprime members of the sequence, then x*y is in the sequence.
Contains all members of A133028 except 3. (End)

			sigma(14)=3*8=24, phi(14)=14*(1/2)*(6/7)=6, sigma(14)/phi(14)=2^2, so 14 is in the list.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 349 terms from Robert Israel)
J. A. B. Dris and C. Leibovici, Why is this sequence not in the OEIS?, October 8 2017.
ProofWiki, Integers whose ratio between sigma and phi is square, misses the second term, 14 as of Dec 2017.

Cf. A000010, A000203, A011257, A020492, A133028, A289336, A289412, A292422.

for n from 1 to 100000 do
    r := numtheory[sigma](n)/numtheory[phi](n) ;
    if issqr(r) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 07 2017

Select[Range[10^5], IntegerQ@ Sqrt[DivisorSigma[1, #]/EulerPhi[#]] &] (* Michael De Vlieger, Dec 08 2017 *)

isok(n) = my(q=sigma(n)/eulerphi(n)); issquare(q) && (denominator(q) == 1); \\ Michel Marcus, Dec 07 2017; corrected Sep 21 2019

a(n) = sigma(n)/phi(n) = m^2, for some integer m.

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)));}

A291185 a(n) = the smallest number k such that sigma(k) / phi(k) >= n.

Original entry on oeis.org

1, 2, 2, 6, 6, 6, 12, 30, 30, 60, 120, 210, 420, 420, 840, 2520, 9240, 9240, 27720, 55440, 120120, 360360, 720720, 2162160, 6126120, 12252240, 36756720, 116396280, 232792560, 698377680, 2677114440, 5354228880, 26771144400, 155272637520, 465817912560

Offset: 1

Jaroslav Krizek, Aug 19 2017

nonn

a(n) = the smallest number k such that A000203(k) / A000010(k) = A289336(k) / A289412(k) >= n.

			For n = 4; a(4) = 6 because 6 is the smallest number with sigma(6) / phi(6) = 12 / 2 = 6 >= 2.

Cf. A000010, A000203, A055234, A289336, A289412.

a:=1; S:=[a]; for n in [2..24] do k:=0; flag:= true; while flag do k+:=1; if &+[d: d in Divisors(k)] / EulerPhi(k) ge n then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;

b:= 0:
for n from 1 to 3*10^6 do
r:= floor(numtheory:-sigma(n)/numtheory:-phi(n));
if r > b then
    for i from b+1 to r do A[i]:= n od:
    b:= r;
fi
od:
seq(A[i],i=1..b); # Robert Israel, Aug 21 2017

With[{s = KeySort@ PositionIndex@ Array[Floor[DivisorSigma[1, #]/EulerPhi@ #] &, 10^6]}, Function[t, Reverse@ FoldList[Min, #] &@ Reverse@ TakeWhile[#, # > 0 &] &@ ReplacePart[t, Map[# -> Lookup[s, #][[1]] &, Keys@ s]]]@ ConstantArray[0, Max@ Keys@ s]] (* Michael De Vlieger, Aug 19 2017 *) (* or *)
r = 1; Reap[ Do[z = DivisorSigma[1, n]/EulerPhi@ n; While[z >= r, r++; Sow@ n], {n, 10^6}]][[2, 1]] (* Giovanni Resta, Aug 21 2017 *)

a(25)-a(35) from Giovanni Resta, Aug 21 2017

cp's OEIS Frontend

A289412 a(n) = denominator of (sigma(n) / phi(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A293391 Integers n such that sigma(n)/phi(n) is a perfect square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A291185 a(n) = the smallest number k such that sigma(k) / phi(k) >= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

Extensions