A055234 -id:A055234 - cp's OEIS frontend

A088830 a(n) = Min{x : sigma(x) = nphi(x), x is not a prime}, the least nonprime solutions to sigma(x) = nphi(x); special balanced numbers.

1, 35, 15, 14, 56, 6, 12, 42, 30, 168, 2580, 210, 630, 420, 840, 20790, 416640, 9240, 291060, 83160, 120120, 5165160, 1719277560, 43825320, 26860680, 277560360, 1304863560, 569729160, 587133466920, 16522145640, 33044291280, 563462139240, 1140028049160

Offset: 1

Labos Elemer, Nov 03 2003

nonn

a(33) > 10^12. - Donovan Johnson, Sep 03 2013

a(34) <= 9015394227840, a(35) <= 1255683068640. - Giovanni Resta, May 08 2017

Amiram Eldar, Table of n, a(n) for n = 1..36 (calculated using data from Jud McCranie)

Compare A087979, which has a slightly different definition.
Cf. A074891, A000203, A000010, A087979, A020492.
Cf. A055234.

ds[x_, de_] := DivisorSigma[1, x]-de*EulerPhi[x] a[n_] := Block[{m=1, s=ds[m, n]}, While[(s !=0||PrimeQ[m])&&!Greater[m, 100000], m++ ]; m]; Table[a[n], {n, 22}]

For n > 3, a(n) = A055234(n). - David Wasserman, Aug 18 2005

More terms from David Wasserman, Aug 18 2005
a(32) from Donovan Johnson, Sep 03 2013
a(33) from Giovanni Resta, May 08 2017

A289336 a(n) = numerator of (sigma(n) / phi(n)).

Original entry on oeis.org

1, 3, 2, 7, 3, 6, 4, 15, 13, 9, 6, 7, 7, 4, 3, 31, 9, 13, 10, 21, 8, 18, 12, 15, 31, 7, 20, 14, 15, 9, 16, 63, 12, 27, 2, 91, 19, 10, 7, 45, 21, 8, 22, 21, 13, 36, 24, 31, 19, 93, 9, 49, 27, 20, 9, 5, 20, 45, 30, 21, 31, 16, 26, 127, 7, 36, 34, 63, 24, 6, 36

Offset: 1

Jaroslav Krizek, Aug 19 2017

			Fractions begin with: 1, 3, 2, 7/2, 3/2, 6, 4/3, 15/4, 13/6, 9/2, 6/5, 7, ...
For n = 7, sigma(7) / phi(7) = 8/6 = 4/3, a(7) = 4.

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. A289412 (denominators).

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

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

Array[Numerator[DivisorSigma[1, #]/EulerPhi[#]] &, 71] (* Michael De Vlieger, Aug 19 2017 *)

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

a(n) = numerator of (A000203(n) / A000010(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A289412(k) = (Pi^4/36) * Product_{p prime} (1 + 2/p^3 - 1/p^5) = 3.6174451656... . - Amiram Eldar, Nov 21 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} A289412(k)/a(k) = 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... (De Koninck and Luca, 2007). - Amiram Eldar, Feb 27 2024

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

Original entry on oeis.org

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)).

A076375 Numbers k such that both k and 2*k are balanced numbers (A020492).

Original entry on oeis.org

1, 3, 6, 15, 35, 70, 105, 210, 357, 420, 1045, 1485, 2090, 2970, 3135, 3339, 5049, 5940, 6270, 10659, 12441, 12540, 16065, 24871, 24969, 29029, 33915, 35343, 39105, 39585, 49742, 50065, 58058, 58435, 64285, 70686, 71145, 74613, 78210, 87087

Offset: 1

Labos Elemer, Oct 15 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)

Cf. A000010, A000203, A055234.

f[x_] := DivisorSigma[1, x]/EulerPhi[x] Do[s=f[n]; s1=f[2*n]; If[IntegerQ[s]&&IntegerQ[s1], Print[n]], {n, 1, 100000}]

A076376 Numbers k such that k, 2k and 4k are balanced numbers (A020492).

Original entry on oeis.org

3, 35, 105, 210, 1045, 1485, 2970, 3135, 6270, 24871, 29029, 35343, 39105, 50065, 58435, 64285, 70686, 71145, 74613, 78210, 87087, 87685, 124605, 137885, 140335, 142290, 149226, 150195, 174174, 175305, 176715, 192855, 249210, 263055, 300390, 350610, 373065

Offset: 1

Labos Elemer, Oct 15 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)

Cf. A000010, A000203, A055234, A020492, A076375.

f[x_] := DivisorSigma[1, x]/EulerPhi[x] Do[s=f[n]; s1=f[2*n]; s2=f[4*n]; If[IntegerQ[s]&&IntegerQ[s1]&&IntegerQ[s2], Print[n]], {n, 1, 1000000}]

A064374 Numbers k such that sigma(k) > phi(k)^2.

Original entry on oeis.org

2, 4, 6, 10, 12, 18, 30

Offset: 1

Labos Elemer, Sep 27 2001

Also numbers k such that sigma(k) > tau(k)*phi(k). - Benoit Cloitre, Aug 06 2002

There are no further terms. - Benoit Cloitre, Aug 06 2002

			k = {2, 4, 6, 10, 12, 18, 30};
sigma(k) = {3, 7, 12, 18, 28, 39, 72};
phi(k) = {1, 2, 2, 4, 4, 6, 8};
phi(k)^2 = {1, 4, 4, 16, 16, 36, 64};
sigma(k) - phi(k)^2 = {2, 3, 8, 2, 12, 3, 8}.
No more solutions below 10000000.

Cf. A000010, A000203, A055234.

Select[Range[30],DivisorSigma[1,#]>EulerPhi[#]^2&] (* Harvey P. Dale, Sep 27 2024 *)

Solutions to A000203(k) > A000010(k)^2.

A064375 Numbers n such that sigma_2(n) > phi(n)^3.

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 12, 14, 18, 20, 24, 30, 36, 42, 60

Offset: 1

Labos Elemer, Sep 27 2001

This sequence is finite, since by Grönwall's theorem sigma_2(n) <= sigma(n)^2 << (n log log n)^2 but phi(n)^3 >> (n/log log n)^3. - Charles R Greathouse IV, Nov 18 2015

			d-square sums:{5, 10, 21, 50, 85, 130, 210, 250, 455, 546, 850, 1300, 1911, 2500, 5460} phi-cubes:{1, 8, 8, 8, 64, 64, 64, 216, 216, 512, 512, 512, 1728, 1728, 4096} differences:{4, 2, 13, 42, 21, 66, 146, 34, 239, 34, 338, 788, 183, 772, 1364} Sequence is believed to be full.

Cf. A001157, A000010, A055234.

Select[Range[100],DivisorSigma[2,#]>EulerPhi[#]^3&] (* Harvey P. Dale, Feb 19 2013 *)

is(n)=my(f=factor(n)); sigma(f,2)>eulerphi(f)^3 \\ Charles R Greathouse IV, Nov 18 2015

Solutions to A001157(n) > A000010(x)^3.

A076377 Numbers k such that k, 2k, 4k and 8*k are balanced numbers (A020492).

Original entry on oeis.org

105, 1485, 3135, 35343, 39105, 71145, 74613, 87087, 124605, 150195, 175305, 192855, 263055, 413655, 421005, 697851, 930699, 1404765, 1873485, 2471931, 2576115, 2965599, 3281265, 3398625, 3937635, 4172259, 4532625, 4589949, 4975965, 5218521, 5474115

Offset: 1

Labos Elemer, Oct 15 2002

nonn

The quotients q = Sigma(u)/phi(u) for u = {n, 2n, 4n, 8n} are integers and for all terms, and equal 4, 12, 14, 15 respectively. For u = 16n, q = 31/2, i.e. no integer was found for u < 6000000.

The comment above is true for terms up to a(238) and true for 985 of the first 1000 terms. - Donovan Johnson, Mar 03 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)

Cf. A000010, A000203, A055234, A020492, A076375, A076376.

f[x_] := DivisorSigma[1, x]/EulerPhi[x] Do[s=f[n]; s1=f[2*n]; s2=f[4*n]; s3=f[8*n] If[IntegerQ[s]&&IntegerQ[s1]&&IntegerQ[s2]&& IntegerQ[s3], Print[n]], {n, 1, 10000000}]

Missing term added by Donovan Johnson, Mar 03 2013

A375262 Least positive integer m such that sigma(m)/phi(m) = n + 1/2, where sigma(.) and phi(.) are given by A000203 and A000010, respectively.

Original entry on oeis.org

5, 459, 4, 10, 860, 18, 24, 11904, 588, 60, 1481172, 1080, 1320, 6236370, 1680, 144480, 10920, 674520, 27720, 25604040, 662535720, 1413720, 303783480, 4324320, 701205120

Offset: 1

Zhi-Wei Sun, Aug 08 2024

Conjecture: Any rational number r >= 1 can be written as sigma(m)/phi(m) with m a positive integer.
We have verified this for rational numbers a/b with 36 >= a >= b >= 1.
In 1977, B.S.K.R. Somayajulu proved that the set {sigma(n)/phi(n): n = 1,2,3,...} is dense in the interval (1,+oo).
a(27) = 790269480. - Chai Wah Wu, Aug 12 2024

			a(1) = 5 with sigma(5)/phi(5) = 6/4 = 1 + 1/2.
a(2) = 459 = 3^3*17 with sigma(459)/phi(459) = 720/288 = 2 + 1/2.
a(20) = 25604040 = 2^3*3*5*7*11*17*163 with sigma(25604040)/phi(25604040) = 102021120/4976640 = 20 + 1/2.

B.S.K.R. Somayajulu, The sequence sigma(n)/phi(n), Math. Student, Vol. 45, No. 1 (1977), pp. 52-54; entire issue.
Zhi-Wei Sun Is it true that {sigma(n)/phi(n): n >= 1} = {r in Q: r >= 1}? Question 476578 at MathOverflow, August 8, 2024.

Cf. A000010, A000203, A055234, A065824.

sigma[n_]:=sigma[n]=DivisorSigma[1,n]; phi[n_]:=phi[n]=EulerPhi[n];
tab={};Do[m=1;While[sigma[m]/phi[m]!=n+1/2,m=m+1];tab=Append[tab,m],{n,1,20}];Print[tab]

a(n) = my(k=1); while (sigma(k)/eulerphi(k) != n + 1/2, k++); k; \\ Michel Marcus, Aug 08 2024

from itertools import count
from math import prod
from sympy import factorint
def A375262(n):
    for m in count(1):
        f = factorint(m)
        if ((n<<1)+1)*m*prod((p-1)**2 for p in f)==prod(p**(e+2)-p for p,e in f.items())<<1:
            return m # Chai Wah Wu, Aug 11 2024

a(21)-a(24) from Amiram Eldar, Aug 08 2024

a(25) from Chai Wah Wu, Aug 12 2024

A291051 a(n) is the smallest number k such that psi(k) = n*phi(k) where psi(k) is Dedekind psi function (A001615) and phi(k) is Euler totient function (A000010), or 0 if no such k exists.

Original entry on oeis.org

1, 3, 2, 14, 190, 6, 78, 42, 30, 570, 16770, 210, 1102290, 2730, 67830, 43890, 133707210, 746130, 27606810, 16546530, 9699690, 417086670, 3828438543930, 8720021310, 705196562070

Offset: 1

Altug Alkan, Aug 17 2017

nonn

Also a(n) is the smallest squarefree number k such that sigma(k) = n*phi(k), or 0 if no such k exists.
It is conjectured that A055234(n) > 0 for each n. Is a(n) > 0 for all values of n?
10^12 < a(26) <= 50353622409090. - Giovanni Resta, Aug 18 2017

			a(4) = 14 since psi(14) / phi(14) = 24 / 6 = 4 and 14 is the least number with this property.

Cf. A000010, A001615, A055234.

psi[n_] := n*Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]; f[n_] := Block[{k = 1}, While[ n*EulerPhi[k] != psi[k], k++]; k]; Array[f, 22] (* Robert G. Wilson v, Sep 15 2017 *)

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
a(n) = {my(k = 1); while (n*eulerphi(k) != a001615(k), k++); k; } \\ Altug Alkan, Aug 17 2017, after Charles R Greathouse IV at A001615

a(23)-a(25) from Giovanni Resta, Aug 18 2017

cp's OEIS Frontend

A088830 a(n) = Min{x : sigma(x) = n*phi(x), x is not a prime}, the least nonprime solutions to sigma(x) = n*phi(x); special balanced numbers.

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A289336 a(n) = numerator of (sigma(n) / phi(n)).

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

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A076375 Numbers k such that both k and 2*k are balanced numbers (A020492).

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A076376 Numbers k such that k, 2*k and 4*k are balanced numbers (A020492).

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A064374 Numbers k such that sigma(k) > phi(k)^2.

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A064375 Numbers n such that sigma_2(n) > phi(n)^3.

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A076377 Numbers k such that k, 2*k, 4*k and 8*k are balanced numbers (A020492).

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A088830 a(n) = Min{x : sigma(x) = nphi(x), x is not a prime}, the least nonprime solutions to sigma(x) = nphi(x); special balanced numbers.

A076376 Numbers k such that k, 2k and 4k are balanced numbers (A020492).

A076377 Numbers k such that k, 2k, 4k and 8*k are balanced numbers (A020492).