A023897 -id:A023897 - cp's OEIS frontend

A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1254, 1485, 1672, 1848, 2090, 2214, 2376, 2436, 2580, 2730, 2970, 3080, 3135, 3339, 3596, 3720, 3828

Offset: 1

David W. Wilson

nonn

The quotient A020492(n)/A002088(n) = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^4/36 or zeta(2)^2 [~2.705808084277845]. - Labos Elemer, Sep 20 2004, corrected by Charles R Greathouse IV, Jun 20 2012
If 2^p-1 is prime (a Mersenne prime) then m = 2^(p-2)*(2^p-1) is in the sequence because when p = 2 we get m = 3 and phi(3) divides sigma(3) and for p > 2, phi(m) = 2^(p-2)*(2^(p-1)-1); sigma(m) = (2^(p-1)-1)*2^p hence sigma(m)/phi(m) = 4 is an integer. So for each n, A133028(n) = 2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence. - Farideh Firoozbakht, Nov 28 2005
Phi and sigma are both multiplicative functions and for this reason if m and n are coprime and included in this sequence then m*n is also in this sequence. - Enrique Pérez Herrero, Sep 05 2010
The quotients sigma(n)/phi(n) are in A023897. - Bernard Schott, Jun 06 2017
There are 544768 balanced numbers < 10^14. - Jud McCranie, Sep 10 2017
a(975807) = 419998185095132. - Jud McCranie, Nov 28 2017

			sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term.

D. Chiang, "N's for which phi(N) divides sigma(N)", Mathematical Buds, Chap. VI pp. 53-70 Vol. 3 Ed. H. D. Ruderman, Mu Alpha Theta 1984.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Jud McCranie, 670314 balanced numbers (first 1000 from T. D. Noe, first 10000 from Donovan Johnson)

Cf. A000010, A000043, A000203, A000668, A011257, A023897, A133028, A291565, A291566, A292422, A351114 (characteristic function).

Positions of 0's in A063514.

[ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; // Klaus Brockhaus, Nov 09 2008

Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]
(* Second program: *)
Select[Range@ 4000, Divisible[DivisorSigma[1, #], EulerPhi@ #] &] (* Michael De Vlieger, Nov 28 2017 *)

select(n->sigma(n)%eulerphi(n)==0,vector(10^4,i,i)) \\ Charles R Greathouse IV, Jun 20 2012

from sympy import totient, divisor_sigma
print([n for n in range(1, 4001) if divisor_sigma(n)%totient(n)==0]) # Indranil Ghosh, Jul 06 2017

from math import prod
from itertools import count, islice
from sympy import factorint
def A020492_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        f = factorint(m)
        if not prod(p**(e+2)-p for p,e in f.items())%(m*prod((p-1)**2 for p in f)):
            yield m
A020492_list = list(islice(A020492_gen(),20)) # Chai Wah Wu, Aug 12 2024

More terms from Farideh Firoozbakht, Nov 28 2005

A055234 Smallest x such that sigma(x) = n*phi(x), or -1 if no such x exists.

Original entry on oeis.org

1, 3, 2, 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, 9015394227840, 1255683068640, 65361608151840

Offset: 1

Jud McCranie, Jun 21 2000

nonn

Conjecture: For each n, a(n) > 0. - Farideh Firoozbakht, Sep 12 2004
a(33) > 10^12. - Donovan Johnson, Mar 06 2012
a(34) <= 9015394227840, a(35) <= 1255683068640. - Giovanni Resta, May 08 2017
Terms after a(36) are > 10^14. a(37) <= 4771397395084320, a(38) <= 2418379501618080, a(39) <= 413956851628320, a(40) <= 1241870554884960, and a(42) <= 50916692750283360. - Jud McCranie, Sep 13 2017
a(38) = 299761858075680, a(39) = 413956851628320. a(37), a(40), and higher terms are > 4.2*10^14. - Jud McCranie, Nov 27 2017
a(37), a(40), and higher terms are > 6.0 x 10^14. - Jud McCranie, Dec 27 2017

			sigma(14) = 24 = 4*phi(14), so a(4) = 14.
n = 21: a(21) = 120120 = 2*2*2*3*5*7*11*13, sigma(120120) = 483840 = n*phi(120120), phi(120120) = 23040.

Cf. A000010, A000203, A020492, A023897, A088830.

a[n_]:=(For[m=1,DivisorSigma[1,m]!=n EulerPhi[m],m++ ];m);Do[Print[a[n]], {n,31}] (* Farideh Firoozbakht, Oct 31 2008 *)

a(n) = {k = 1; while(sigma(k) != n*eulerphi(k), k++); k;} \\ Michel Marcus, Sep 01 2014

from math import prod
from itertools import count
from sympy import factorint
def A055234(n):
    for m in count(1):
        f = factorint(m)
        if n*m*prod((p-1)**2 for p in f)==prod(p**(e+2)-p for p,e in f.items()):
            return m # Chai Wah Wu, Aug 12 2024

a(n) = Min{x : A000203(x)/A000010(x) = n} = Min{x : A023897(x) = n}

More terms from Farideh Firoozbakht, Sep 12 2004
a(32) from Donovan Johnson, Mar 06 2012
a(33) from Giovanni Resta, May 08 2017
a(34)-a(36) from Jud McCranie, Sep 10 2017

A063514 a(n) = sigma(n) mod phi(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 2, 3, 1, 2, 2, 0, 2, 0, 0, 7, 2, 3, 2, 2, 8, 6, 2, 4, 11, 6, 4, 8, 2, 0, 2, 15, 8, 6, 0, 7, 2, 6, 8, 10, 2, 0, 2, 4, 6, 6, 2, 12, 15, 13, 8, 2, 2, 12, 32, 0, 8, 6, 2, 8, 2, 6, 32, 31, 36, 4, 2, 30, 8, 0, 2, 3, 2, 6, 4, 32, 36, 0, 2, 26, 13, 6, 2

Offset: 1

Labos Elemer, Jul 31 2001

If a(n) = 0, then n is a balanced number (A020492).

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A000010, A000203, A020492, A023897.

[SumOfDivisors(n) mod EulerPhi(n): n in [1..85]]; // Bruno Berselli, Jan 31 2013

a[n_] := Mod[DivisorSigma[1, n], EulerPhi[n]]; Array[a, 100] (* Amiram Eldar, Dec 25 2024 *)

a(n) = { sigma(n)%eulerphi(n) } \\ Harry J. Smith, Aug 24 2009

a(p^2) = 2*p+1, for prime p >= 5. - Michel Marcus, Apr 07 2020

A351112 Number of balanced numbers dividing n.

Original entry on oeis.org

1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 5, 1, 3, 3, 2, 1, 4, 1, 2, 2, 2, 1, 5, 1, 2, 2, 3, 1, 6, 1, 2, 2, 2, 2, 5, 1, 2, 2, 2, 1, 6, 1, 2, 3, 2, 1, 5, 1, 2, 2, 2, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 2, 2, 1, 4, 1, 2, 2, 5, 1, 5, 1, 2, 3, 2, 1, 5, 1, 2, 2, 2, 1, 7, 1, 2, 2, 2, 1, 6, 1, 2

Offset: 1

Wesley Ivan Hurt, Jan 31 2022

nonn

A balanced number k is a number such that phi(k) | sigma(k).

Inverse Möbius transform of the characteristic function of balanced numbers (A351114). - Wesley Ivan Hurt, Jun 23 2024

			a(4) = 2; the balanced divisors of 4 are 1 and 2.
a(5) = 1; 1 is the only balanced divisor of 5.
a(6) = 4; the balanced divisors of 6 are 1,2,3,6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A351113 (sum of the balanced numbers dividing n).

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A023897, A351114.

f:= proc(n) uses numtheory;
  nops(select(t -> sigma(t) mod phi(t) = 0, divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Nov 28 2023

a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], EulerPhi[#]] &]; Array[a, 100] (* Amiram Eldar, Feb 01 2022 *)

a(n) = sumdiv(n, d, if (!(sigma(d) % eulerphi(d)), 1)); \\ Michel Marcus, Feb 01 2022

a(n) = Sum_{d|n, phi(d)|sigma(d)} 1.
a(n) = Sum_{d|n} A351114(d).
a(n) = tau(n) - Sum_{d|n} sign(sigma(d) mod phi(d)).
Conjecture: asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A020492(k) = 2.4343... (assuming empirically that this sum of reciprocals converges). - Amiram Eldar, Dec 27 2024

A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

Original entry on oeis.org

1, 3, 18, 12, 200, 14, 3364, 15, 722, 328, 9801, 42, 25281, 116, 1800, 165, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 172, 98942809, 488, 1547536, 336, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 319, 3150464641

Offset: 1

Labos Elemer, Nov 12 2002

nonn

Values are frequently identical to terms of A077102. Since gcd(a,b) and gcd(a+b,a-b) may differ, so may the smallest solutions. A077102(m) and a(m) differ at m = 1, 2, 4, 8, 16, 28, 32, 40, etc.

Giovanni Resta, Table of n, a(n) for n = 1..120

Cf. A000203, A000010, A009223, A055008, A077099-A077102, A051612, A065387, A023897, A020492.

f[x_] := Apply[GCD, {DivisorSigma[1, x], EulerPhi[x]}] t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10^13}];

a(n)=my(x=n);while(gcd(eulerphi(x),sigma(x))!=n, x++); x \\ Charles R Greathouse IV, Dec 09 2013

a(n) = Min{x; A055008(x)=n}. a(n)=Min{x; gcd(A000203(x), A000010(x))=n}

a(n) = Min{x: A023897(x)= n}, smallest balanced number (A020492) for which the quotient equals n.

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)).

A342103 Balanced numbers (A020492) that are also arithmetic numbers (A003601).

Original entry on oeis.org

1, 3, 6, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1254, 1485, 1672, 1848, 2090, 2214, 2376, 2436, 2580, 2730, 2970, 3080, 3135, 3339, 3596, 3720, 3828, 3956, 4064, 4180

Offset: 1

Bernard Schott, Feb 28 2021

nonn

Equivalently, numbers m such that phi(m) (A000010) and tau(m) (A000005) both divide sigma(m) (A000203). In this case, the quotients sigma(m)/phi(m) = A023897(m) and sigma(m)/tau(m) = A102187(m).
Phi, tau and sigma are multiplicative functions and for this reason if k and q are coprime and included in this sequence then k*q is another term.
The only prime in the sequence is 3, because sigma(2)/tau(2) = 3/2 and when p is an odd prime, sigma(p)/phi(p) = (p+1)/(p-1) is an integer iff p=3 with sigma(3)/phi(3) = 4/2 = 2, and also sigma(3)/tau(3) = 4/2 = 2.

			phi(30) = tau(30) = 8, sigma(30) = 72 and 72/8 = 9, hence 30 is a term.
phi(12) = 4, tau(12) = 6, sigma(12) = 28, phi(12) divides sigma(12), but tau(12) does not divide sigma(12), hence 12 is a balanced number but is not an arithmetic number, and 12 is not a term.
phi(20) = 8, tau(20) = 6, sigma(20) = 42, tau(20) divides sigma(20), but phi(20) does not divide sigma(20), hence 20 is an arithmetic number but is not a balanced number, and 20 is not a term.

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

Intersection of A003601 and A020492.
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A023897 (sigma/phi), A102187 (sigma/tau).
Cf. A342104, A342105, A342106.

with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) = 0) : select(filter, [$1..5000]);

Select[Range[5000], And @@ Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] &] (* Amiram Eldar, Feb 28 2021 *)

isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && !(s % numdiv(m)); \\ Michel Marcus, Mar 01 2021

A351113 Sum of the balanced numbers dividing n.

Original entry on oeis.org

1, 3, 4, 3, 1, 12, 1, 3, 4, 3, 1, 24, 1, 17, 19, 3, 1, 12, 1, 3, 4, 3, 1, 24, 1, 3, 4, 17, 1, 57, 1, 3, 4, 3, 36, 24, 1, 3, 4, 3, 1, 68, 1, 3, 19, 3, 1, 24, 1, 3, 4, 3, 1, 12, 1, 73, 4, 3, 1, 69, 1, 3, 4, 3, 1, 12, 1, 3, 4, 122, 1, 24, 1, 3, 19, 3, 1, 90, 1, 3, 4, 3, 1, 80

Offset: 1

Wesley Ivan Hurt, Jan 31 2022

nonn

A balanced number k is a number such that phi(k) | sigma(k).

			a(4) = 3; the balanced divisors of 4 are 1 and 2 and 1+2 = 3.
a(5) = 1; 1 is the only balanced divisor of 5.
a(6) = 12; the balanced divisors of 6 are 1,2,3,6 and 1+2+3+6 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A351112 (number of balanced divisors of n).

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A023897, A351114.

a[n_] := DivisorSum[n, # &, Divisible[DivisorSigma[1, #], EulerPhi[#]] &]; Array[a, 100] (* Amiram Eldar, Feb 01 2022 *)

a(n) = sumdiv(n, d, if (!(sigma(d) % eulerphi(d)), d)); \\ Michel Marcus, Feb 01 2022

a(n) = Sum_{d|n, phi(d)|sigma(d)} d.
a(n) = Sum_{d|n} d * A351114(d).
a(n) = sigma(n) - Sum_{d|n} d * sign(sigma(d) mod phi(d)).

A342104 Balanced numbers (A020492) that are not arithmetic numbers (A003601).

Original entry on oeis.org

2, 12, 18630, 27000, 443394, 6242022, 14412720, 22315419, 26744100, 44630838, 50496960, 106034880, 128710944, 148536990, 162907584, 212072880, 218470770, 296259930, 349444530, 397253968, 535267776, 641250900, 641418960, 666274653, 684165552, 688208724, 709639408

Offset: 1

Bernard Schott, Feb 28 2021

nonn

Equivalently, numbers m such that phi(m) divides sigma(m) but tau(m) does not divide sigma(m), the corresponding quotients sigma(m)/phi(m) = A023897(m).

The only prime in the sequence is 2, because sigma(2)/phi(2) = 3 and sigma(2)/tau(2) = 3/2; then, if p odd prime, sigma(p)/phi(p) = (p+1)/(p-1) is an integer iff p = 3, but for p = 3, tau(3) divides sigma(3) with sigma(3)/tau(3) = 4/2 = 2.

			Sigma(12) = 28, phi(12) = 4 and tau(12) = 6, hence phi(12) divides sigma(12), but tau(12) does not divide sigma(12), so 12 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..5620

Equals A020492 \ A003601.
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A023897 (sigma/phi).
Cf. A342103, A342105, A342106.

with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) <> 0) : select(filter, [$1..500000]);

Select[Range[500000], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {False, True} &] (* Amiram Eldar, Feb 28 2021 *)

isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && (s % numdiv(m)); \\ Michel Marcus, Mar 01 2021

a(5)-a(27) from Amiram Eldar, Feb 28 2021

cp's OEIS Frontend

A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

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A055234 Smallest x such that sigma(x) = n*phi(x), or -1 if no such x exists.

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A063514 a(n) = sigma(n) mod phi(n).

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A351112 Number of balanced numbers dividing n.

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A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

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

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Magma

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

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