A011257 -id:A011257 - 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

A163667 Numbers n such that sigma(n) = 9*phi(n).

Original entry on oeis.org

30, 264, 714, 3080, 3828, 6678, 10098, 12648, 21318, 22152, 24882, 44660, 49938, 61344, 86304, 94944, 118296, 129504, 130356, 147560, 183396, 199386, 201756, 207264, 216936, 248710, 258440, 265914, 275196, 290290, 321204, 505164, 628776, 706266, 706836

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Aug 09 2009

This sequence is a subsequence of A011257 because sqrt(phi(n)*sigma(n)) = 3*phi(n).

If 2^p-1 and 2*3^k-1 are two primes greater than 5 then n = 2^(p-2)*(2^p-1)*3^(k-1)*(2*3^k-1) (the product of two relatively prime terms 2^(p-2)*(2^p-1) and 3^(k-1)*(2*3^k-1) of A011257) is in the sequence. The proof is easy.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

Cf. A000010, A000043, A000203, A000668, A003307, A011257, A079363.

Select[Range[700000],DivisorSigma[1,# ]==9EulerPhi[ # ]&]

is(n)=sigma(n)==9*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

A164646 Numbers n such that sigma(n)/phi(n) = 9/4.

Original entry on oeis.org

51, 477, 595, 3567, 17765, 20735, 41615, 104931, 276651, 470721, 493493, 599169, 834591, 993395, 1092845, 1242505, 1318521, 1479981, 1490645, 1712037, 2344045, 2736305, 2912463, 2986941, 2990709, 3042873, 3187917, 3277611, 3295821, 3767331, 4686039, 5059881

Offset: 1

M. F. Hasler, Aug 22 2009

nonn

A subsequence of A011257.
If 3^{k+1}-1 = d*D such that p = 2*b^{k+1}*(d+1) - 1 and q = 2*(b^{k+1}+D)-1 are distinct primes, then n = 3^k*p*q is a term of this sequence.
The same theorem holds for sequences of numbers such that sigma/phi=b^2/(b-1)^2 with other primes b (here b=3; in A068390: b=2, in A164648: b=5).

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

Cf. A000010 (=phi), A000203 (=sigma), A068390 (sigma/phi=4), A163667 (sigma/phi=9), A164647 (sigma/phi=16/9).

Select[Range[506*10^4],DivisorSigma[1,#]/EulerPhi[#]==9/4&] (* Harvey P. Dale, Jun 22 2019 *)

for( n=1,1e7, sigma(n)==9/4*eulerphi(n) && print1(n","))

A164648 Numbers k such that sigma(k)/phi(k) = 25/16.

Original entry on oeis.org

40859, 48505, 54385, 121771, 156125, 565607, 1154419, 1219933, 1294363, 2448397, 3590461, 9710975, 16067363, 16069573, 17984515, 19013455, 21341755, 25804115, 26515223, 27656155, 29655415, 30372605, 32101255, 34467653, 36546355, 38043943, 38645981, 39559219

Offset: 1

M. F. Hasler, Aug 22 2009

A subsequence of A011257.
If 5^{k+1}-1 = d*D such that p = 2*5^{k+1}*(d+1)-1 and q = 2*(5^{k+1}+D)-1 are distinct primes, then n = 5^k*p*q is a term of this sequence.
The same theorem holds for sequences of numbers such that sigma/phi=b^2/(b-1)^2 with other primes b (here b=5), cf. A164646.

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

Cf. A000010 (=phi), A000203 (=sigma), A068390 (sigma/phi=4), A163667 (sigma/phi=9), A164646 (sigma/phi=9/4).

Select[Range[2000000], DivisorSigma[1, #]/EulerPhi[#] == 25/16 &] (* Carl Najafi, Aug 16 2011 *)

for( n=1,1e7, sigma(n)==25/16*eulerphi(n) && print1(n","))

More terms from Carl Najafi, Aug 16 2011

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.

A065146 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and sigma(n) are all integers.

Original entry on oeis.org

1, 248, 264, 418, 477, 1485, 3080, 3135, 3596, 3828, 5396, 10098, 12648, 20026, 21318, 22152, 23374, 24882, 35074, 35343, 39105, 41656, 44660, 49938, 55154, 56536, 61344, 71145, 74613, 86304, 87087, 104931, 118296, 124605, 129504, 130356, 147560, 150195

Offset: 1

Labos Elemer, Oct 18 2001

nonn

			n = 5396, phi(n) = 2520, sigma(n) = 10080, arithmetic mean = 6300, geometric mean = 5040, harmonic mean = 4032; 67 cases < 10^6.

Harry J. Smith and Donovan Johnson, Table of n, a(n) for n = 1..5000 (first 500 terms from Harry J. Smith)

Cf. A000010, A000203, A062354, A011257.

Do[s = EulerPhi[n]*DivisorSigma[1, n]; z = (EulerPhi[n]+DivisorSigma[1, n])/2; u = h[n]; If[IntegerQ[Sqrt[s]]&&IntegerQ[z]&&IntegerQ[u], Print[n]], {n, 1, 1000000}]

{ n=0; for (m=1, 10^9, e=eulerphi(m); s=sigma(m); if (!issquare(e*s), next); h=(2*e*s)/(e + s); if (frac(h) != 0, next); if (frac((e + s)/2) != 0, next); write("b065146.txt", n++, " ", m); if (n==500, return) ) } \\ Harry J. Smith, Oct 12 2009

a = (phi(n)+sigma(n))/2, g = sqrt(phi(n)*sigma(n)), h = (2*phi(n)*sigma(n))/(phi(n)+sigma(n)) = g^2/a are all integers; phi() = A000010(), sigma() = A000203().

A164647 Numbers n such that sigma(n)/phi(n) = 16/9.

Original entry on oeis.org

1463, 2945, 8255, 70091, 81809, 89999, 122759, 187625, 193039, 196469, 388585, 494665, 671365, 2311673, 2442583, 2687113, 4209985, 4705285, 4902247, 5393017, 5667389, 5866003, 9248323, 10795967, 11345411, 11670275, 11773027, 13290485, 13741273, 15978487

Offset: 1

M. F. Hasler, Aug 22 2009

nonn

A subsequence of A011257.

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

Cf. A000010 (=phi), A000203 (=sigma), A068390, A163667, A164646.

for( n=1,10^7, sigma(n)==16/9*eulerphi(n) && print1(n","))

More terms from Farideh Firoozbakht, Sep 22 2009

A327624 Numbers m such that sigma(m)*phi(m) is a square but sigma(m)/phi(m) is not an integer.

Original entry on oeis.org

51, 170, 194, 364, 405, 477, 595, 679, 760, 780, 1023, 1455, 1463, 1496, 1512, 1524, 1674, 1715, 1731, 1796, 1804, 2058, 2080, 2651, 2754, 2945, 3192, 3410, 3534, 3567, 4381, 4420, 5044, 5130, 5670, 5770, 5784, 5797, 5822, 5859, 7600, 8245

Offset: 1

Bernard Schott, Sep 19 2019

nonn

If sigma(m)/phi(m) is a square (m is in A293391) then sigma(m)*phi(m) is also a square (m is in A011257), but the converse is false (see 51 in the Example section). This sequence consists of these counterexamples.

			phi(51) = 32 and sigma(51) = 72, phi(51) * sigma(51) = 32 * 72 = 2304 = 48^2, but sigma(51)/phi(51) = 72/32 = 9/4 is not an integer.

Equals A293391 \ A011257.
Cf. A020492 (sigma(m)/phi(m) is an integer).
Cf. A000010 (phi), A000203 (sigma).

[k:k in [1..9000]| not IsIntegral(SumOfDivisors(k)/EulerPhi(k)) and IsSquare(EulerPhi(k)*SumOfDivisors(k)) ]; // Marius A. Burtea, Sep 19 2019

filter:= v -> sigma(v)/phi(v) <> floor(sigma(v)/phi(v)) and issqr(sigma(v)*phi(v)) : select(filter, [$1..50000]);

sQ[n_] := IntegerQ @ Sqrt[n]; aQ[n_] := sQ[(p = EulerPhi[n]) * (s = DivisorSigma[1, n])] && !sQ[s/p]; Select[Range[10^4], aQ] (* Amiram Eldar, Sep 19 2019 *)

isok(m) = my(s=sigma(m), e=eulerphi(m)); issquare(s*e) && (s%e); \\ Michel Marcus, Sep 21 2019

A067781 Numbers k such that phi(k) and sigma(k) are both perfect squares.

Original entry on oeis.org

1, 170, 364, 679, 5044, 5130, 5670, 5770, 8721, 8736, 9154, 9639, 9809, 14322, 16376, 22413, 27783, 30256, 32025, 37114, 38760, 51455, 71604, 78570, 82615, 88392, 92004, 100821, 101153, 104168, 115430, 121056, 133569, 139954, 148568, 171069, 177940, 198462, 217868

Offset: 1

Benoit Cloitre, Feb 07 2002

nonn

A subsequence of A011257. - M. F. Hasler, Sep 22 2009

Donovan Johnson, Table of n, a(n) for n = 1..5000

Cf. A000010 (phi), A000203 (sigma), A006532, A011257, A039770.

Select[Range[250000], And @@ (IntegerQ[Sqrt[#1]] & /@ {EulerPhi[#], DivisorSigma[1, #]} ) &] (* Amiram Eldar, May 08 2025 *)

isok(k) = {my(f = factor(k)); issquare(eulerphi(f)) && issquare(sigma(f));} \\ Amiram Eldar, May 08 2025

Equals A006532 intersect A039770. - M. F. Hasler, Sep 22 2009

A114078 Numbers k such that phi(k)*sigma(k) is a fourth power.

Original entry on oeis.org

1, 170, 595, 714, 121056, 480441, 529620, 706063, 706237, 729752, 755972, 815654, 2162808, 2449062, 2827789, 2927848, 2957416, 2994681, 2995419, 3010227, 3019028, 3019772, 3080140, 3093860, 3103464, 3206364, 3213804

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			phi(595) * sigma(595) = 331776 = 24^4.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 170, p. 54, Ellipses, Paris 2008.

Amiram Eldar, Table of n, a(n) for n = 1..600

Cf. A062354, subsequence of A011257.

[k:k in [1..3300000]| IsPower(EulerPhi(k)*DivisorSigma(1,k),4)]; // Marius A. Burtea, Sep 19 2019

Select[Range[3300000],IntegerQ[Power[EulerPhi[#] DivisorSigma[1,#], (4)^-1]]&]  (* Harvey P. Dale, Mar 14 2011 *)

isok(n) = ispower(eulerphi(n)*sigma(n), 4); \\ Michel Marcus, Jan 09 2014

cp's OEIS Frontend

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

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A163667 Numbers n such that sigma(n) = 9*phi(n).

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A164646 Numbers n such that sigma(n)/phi(n) = 9/4.

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A164648 Numbers k such that sigma(k)/phi(k) = 25/16.

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

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A065146 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and sigma(n) are all integers.

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A164647 Numbers n such that sigma(n)/phi(n) = 16/9.

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