A062354 -id:A062354 - cp's OEIS frontend

A086768 Number of conjugacy classes in the group GL(3,Z_n).

1, 6, 24, 60, 120, 144, 336, 536, 714, 720, 1320, 1440, 2184, 2016, 2880, 4528, 4896, 4284, 6840, 7200, 8064, 7920, 12144, 12864, 15540, 13104, 19908, 20160, 24360, 17280, 29760, 37216, 31680, 29376, 40320, 42840, 50616, 41040, 52416, 64320, 68880, 48384, 79464, 79200, 85680, 72864

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 02 2003

Cf. A064767, A062354.

[Nclasses(GeneralLinearGroup(3, ResidueClassRing(n))) : n in [2..50]]; // Robin Visser, Aug 06 2023

For a prime p : a(p) = p*(p^2 - 1).

More terms from Robin Visser, Aug 06 2023

A114077 Numbers k such that phi(k)*sigma(k) is a cube.

Original entry on oeis.org

1, 3, 119, 357, 2522, 6305, 6596, 6604, 7566, 18915, 19788, 19812, 20520, 24347, 24353, 26068, 26082, 28126, 47959, 58520, 64790, 65205, 70315, 73041, 73059, 73636, 75460, 78204, 84378, 143877, 156519, 164920, 175560, 194370, 210945, 212660, 220908, 226380

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			phi(357) * sigma(357) = 110592 = 48^3.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 119, p. 41, Ellipses, Paris 2008.

Amiram Eldar, Table of n, a(n) for n = 1..6000

Cf. A062354.

[k:k in [1..230000]| IsPower(EulerPhi(k)*DivisorSigma(1,k),3)]; // Marius A. Burtea, Sep 19 2019

Select[Range[10^5], IntegerQ @ Surd[EulerPhi[#] * DivisorSigma[1, #], 3] &] (* Amiram Eldar, Sep 19 2019 *)

isok(n) = ispower(eulerphi(n)*sigma(n), 3); \\ Michel Marcus, Jan 22 2014

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

A131262 a(n) = least index k such that A130654(k) = n.

Original entry on oeis.org

1, 3, 14, 60, 248, 1008

Offset: 0

Alexander Adamchuk, Jun 24 2007

Also a(n) = least index k such that A092505(k) = A002430(k) / A046990(k) = 2^n.
Note that
a(0) = 1 = 1 - 0 = 2^0 - 0;
a(1) = 3 = 4 - 1 = 2^2 - 1;
a(2) = 14 = 16 - 2 = 2^4 - 2;
a(3) = 60 = 64 - 4 = 2^6 - 4;
a(4) = 248 = 256 - 8 = 2^8 - 8.
Conjecture: a(n) = Sigma(2^n)*EulerPhi(2^n) = 2^(2n) - Floor(2^n/2) = A062354(2^n).
If this conjecture is true the next term would be a(5) = 1008 = 1024 - 16 = 2^10 - 16.

			A130654(n) begins
{0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 0, ...}.
Thus a(0) = 1, a(1) = 3, a(2) = 14, a(3) = 60.

Cf. A130654 = Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n). Cf. A092505 = A002430(n) / A046990(n), n>0. Cf. A002430 = Numerators in Taylor series for tan(x). Cf. A046990 = Numerators of Taylor series for log(1/cos(x)). Cf. A062354 = Sigma(n)*EulerPhi(n).

Conjecture: a(n) = Sigma(2^n)*EulerPhi(2^n) = 2^(2n) - Floor(2^n/2) = A062354(2^n).

a(5) = 1008 from Alexander Adamchuk, May 02 2010

A386401 a(n) = numerator(sigma(n)*phi(n)/n^2).

Original entry on oeis.org

1, 3, 8, 7, 24, 2, 48, 15, 26, 18, 120, 7, 168, 36, 64, 31, 288, 13, 360, 21, 128, 90, 528, 5, 124, 126, 80, 6, 840, 16, 960, 63, 320, 216, 1152, 91, 1368, 270, 448, 9, 1680, 32, 1848, 105, 208, 396, 2208, 31, 342, 93, 256, 147, 2808, 20, 576, 45, 320, 630, 3480, 56

Offset: 1

Stefano Spezia, Jul 20 2025

a(n)/A386402(n) = sigma(n)*phi(n)*(1/n^2) is a multiplicative function since it is the product of three multiplicative functions.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 5.3.21 on page 169.

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

Cf. A000010, A000203, A000290, A062354, A065465.

Cf. A386402 (denominators).

a[n_]:=Numerator[DivisorSigma[1,n]EulerPhi[n]/n^2]; Array[a,60]

a(n) = {my(f = factor(n)); numerator(sigma(f) * eulerphi(f) / n^2);} \\ Amiram Eldar, Jul 21 2025

From Amiram Eldar, Jul 21 2025: (Start)
Let f(n) = a(n)/A386402(n) = sigma(n)*phi(n)/n^2. Then:
f(n) = A062354(n)/n^2.
f(n) is multiplicative with f(p^e) = 1 - 1/p^(e+1).
Dirichlet g.f. of f(n): zeta(s) * zeta(s+1) * Product_{p prime} (1 - 1/p^(s+1) - 1/p^(s+2)+ 1/p^(2*s+2)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465). (End)

A015710 Least k >= 0 such that phi(n) * sigma(n) + k^2 is a perfect square, or -1 if impossible.

Original entry on oeis.org

0, 1, 1, -1, 1, 1, 1, 2, -1, 3, 1, 3, 1, 0, 2, 29, 1, -1, 1, 5, 4, 1, 1, 2, 26, 5, 3, 2, 1, 0, 1, 4, 1, 6, 2, 8, 1, 3, 5, 2, 1, 2, 1, 1, 8, 4, 1, 15, -1, 16, 0, 7, 1, 7, 6, 6, 6, 9, 1, 4, 1, 6, 10, 119, 8, 6, 1, 8, 1, 5, 1, 9, 1, 11, 9, 1, 4, 8, 1, 17, -1, 1

Offset: 1

Robert G. Wilson v

sign

Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A015713 (a(n) is zero), A062354 (phi(n)*sigma(n)).

a[n_] := Module[{m = EulerPhi[n]*DivisorSigma[1, n]}, If[Mod[m, 4] == 2, -1, k = 0; While[!IntegerQ[Sqrt[m + k^2]], k++]; k]]; Array[a, 100] (* Amiram Eldar, Dec 07 2018 *)

a(n) = {my(x = sigma(n)*eulerphi(n)); if ((x % 4) == 2, -1, my(k=0); while (! issquare(x+k^2), k++); k;);} \\ Michel Marcus, Dec 07 2018

a(14), a(30), and a(51) corrected by Sean A. Irvine, Dec 06 2018

Entry revised by Amiram Eldar, Sean A. Irvine, and Michel Marcus, Dec 06 2018

A065148 Nonprimes m such that phi(m)*sigma(m) is divisible by m+1.

Original entry on oeis.org

15, 20, 35, 95, 104, 143, 207, 255, 287, 319, 323, 464, 539, 650, 890, 899, 1023, 1034, 1199, 1295, 1349, 1407, 1519, 1763, 1952, 2015, 2204, 2834, 2975, 3599, 4031, 4454, 4607, 5183, 6479, 9215, 9503, 9799, 10403, 11339, 11663, 12095, 12824, 13055

Offset: 1

Labos Elemer, Oct 18 2001

nonn

Every prime p satisfies A000010(p)*A000203(p) == 0 (mod p+1).

			m = 95 is a term since phi(95) = 72, sigma(95) = 120, product = 8640, product/(m+1) = 90.

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..500 from Harry J. Smith)

Cf. A000010, A000203, A062354, A011257.

Do[s=EulerPhi[n]*DivisorSigma[1, n]; If[IntegerQ[s/(n+1)]&&!PrimeQ[n], Print[n]], {n, 1, 100000}]
Select[Range[14000],!PrimeQ[#]&&Divisible[EulerPhi[#]DivisorSigma[1,#],#+1]&] (* Harvey P. Dale, Jul 08 2017 *)

{ n=0; for (m=1, 10^9, s=eulerphi(m)*sigma(m); if (s%(m+1) == 0 && !isprime(m), write("b065148.txt", n++, " ", m); if (n==500, return)) ) } \\ Harry J. Smith, Oct 12 2009

A000010(m)*A000203(m) == 0 (mod m+1), m is composite.

Offset changed from 0 to 1 by Harry J. Smith, Oct 12 2009

Definition clarified by Harvey P. Dale, Jul 08 2017

A065149 Composite numbers m such that phi(m)*sigma(m) is divisible by m-1.

Original entry on oeis.org

10, 33, 65, 136, 145, 261, 385, 451, 897, 946, 1281, 1441, 1665, 1729, 2241, 2353, 3585, 5185, 6721, 7201, 8380, 8911, 8961, 11521, 11782, 12673, 12801, 17101, 18241, 20737, 25201, 26625, 26677, 26937, 29697, 29953, 30721, 30889, 32896

Offset: 1

Labos Elemer, Oct 18 2001

nonn

			m=136, phi(136)=64, sigma(136)=270, product=17280, quotient=128; for primes the formula holds.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A000010, A000203, A062354, A011257.

Filtered([2..40000],m->Phi(m)*Sigma(m) mod (m-1)=0 and not IsPrime(m)); # Muniru A Asiru, Jun 18 2018

with(numtheory): select(m->modp(phi(m)*sigma(m),m-1)=0 and not isprime(m),[$2..40000]); # Muniru A Asiru, Jun 18 2018

Do[s=EulerPhi[n]*DivisorSigma[1, n]; If[IntegerQ[s/(n-1)]&&!PrimeQ[n], Print[n]], {n, 1, 100000}]

{ n=0; for (m=2, 10^9, s=eulerphi(m)*sigma(m); if (s%(m-1) == 0 && !isprime(m), write("b065149.txt", n++, " ", m); if (n==500, return)) ) } \\ Harry J. Smith, Oct 12 2009

(A000010(m)*A000203(m)) mod (m-1) = 0, m is composite.

Offset changed from 0 to 1 by Harry J. Smith, Oct 12 2009

A065550 a(n) = floor(sqrt(phi(w)*sigma(w)+w^2)), where w=10^n.

Original entry on oeis.org

13, 136, 1391, 14030, 140865, 1411444, 14128309, 141352267, 1413868217, 14140409111, 141412724154, 1414170403052, 14141919829640, 141420277272713, 1414208167563878, 14142108649717545, 141421221367320690, 1414212888023339560, 14142132251982630599, 141421339378569021517

Offset: 1

Labos Elemer, Nov 13 2001

nonn

a(n) tends to sqrt(2)*(10^n) when n->oo.

Robert Israel, Table of n, a(n) for n = 1..995

Cf. A000290, A062354, A065655, A065656.

a:= n -> floor(sqrt(2*100^n - 20^n/5 - 50^n/2 + 10^n/10)):
map(a, [$1..100]); # Robert Israel, Dec 03 2024

a[n_] := Floor[Sqrt[EulerPhi[10^n] * DivisorSigma[1, 10^n] + 100^n]]; Array[a, 20] (* Amiram Eldar, Jun 12 2022 *)

a(n) = my(w=10^n); sqrtint(eulerphi(w)*sigma(w)+w^2); \\ Michel Marcus, Mar 23 2020

from sympy import integer_nthroot, totient as phi, divisor_sigma as sigma
def isqrt(n): return integer_nthroot(n, 2)[0]
def a(n): w = 10**n; return isqrt(phi(w)*sigma(w, 1) + w**2)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 12 2022

a(n) = floor(sqrt(A062354(w) + A000290(w))), where w=10^n.

a(n) = floor(10^n * sqrt(2 - 5^(-n-1) - 2^(-n-1) + 10^(-n-1))). - Robert Israel, Dec 03 2024

Corrected and extended by Michel Marcus, Jun 12 2022

A065558 Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the maximal degree of an irreducible representation of G_n.

Original entry on oeis.org

1, 2, 4, 6, 6, 8, 8, 12, 12, 12, 12, 24, 14, 16, 24, 24, 18, 24, 20, 36, 32, 24, 24, 48

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 29 2001

nonn

a(n) is multiplicative and for an odd prime p : a(p) = p + 1 . The number of irreducible representations of G_n is in sequence A062354.

Conjecture: a(2n) = 2*A001615(n). - Ralf Stephan, Mar 26 2004

A000252, A062354

cp's OEIS Frontend

A086768 Number of conjugacy classes in the group GL(3,Z_n).

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Magma

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A114077 Numbers k such that phi(k)*sigma(k) is a cube.

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Magma

Mathematica

PARI

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

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Magma

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A131262 a(n) = least index k such that A130654(k) = n.

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A386401 a(n) = numerator(sigma(n)*phi(n)/n^2).

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A015710 Least k >= 0 such that phi(n) * sigma(n) + k^2 is a perfect square, or -1 if impossible.

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A065148 Nonprimes m such that phi(m)*sigma(m) is divisible by m+1.

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A065149 Composite numbers m such that phi(m)*sigma(m) is divisible by m-1.

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