A062355 -id:A062355 - cp's OEIS frontend

A062354 a(n) = sigma(n)*phi(n).

1, 3, 8, 14, 24, 24, 48, 60, 78, 72, 120, 112, 168, 144, 192, 248, 288, 234, 360, 336, 384, 360, 528, 480, 620, 504, 720, 672, 840, 576, 960, 1008, 960, 864, 1152, 1092, 1368, 1080, 1344, 1440, 1680, 1152, 1848, 1680, 1872, 1584, 2208, 1984, 2394, 1860

Offset: 1

Jason Earls, Jul 06 2001

Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the number of conjugacy classes in G_n. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 13 2001
a(n) = Sum_{d|n} phi(n*d). - Vladeta Jovovic, Apr 17 2002
Apparently the Mobius transform of A062952. - R. J. Mathar, Oct 01 2011

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.

T. D. Noe, Table of n, a(n) for n=1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^3 / 18) for n = 1..1000000
J.-L. Nicolas and Jonathan Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis, arXiv:1211.6944 [math.HO], 2012, to appear in RAMA125 Proceedings, Contemp. Math.

Cf. A000010, A000203, A000252, A062355, A064840, A093827.

Table[EulerPhi[n] DivisorSigma[1, n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n)=sigma(n)*eulerphi(n); vector(150,n,a(n))

Multiplicative with a(p^e) = p^(e-1)*(p^(e+1)-1). - Vladeta Jovovic, Apr 17 2002
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*product_{primes p} (1-p^(1-s)-p^(-s)+p^(2-2s)). - R. J. Mathar, Oct 01 2011, corrected by Vaclav Kotesovec, Dec 17 2019
6/Pi^2 < a(n)/n^2 < 1 for n > 1. - Jonathan Sondow, Mar 06 2014
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.535896... - Vaclav Kotesovec, Dec 17 2019
Sum_{n>=1} 1/a(n) = 1.7865764... (A093827). - Amiram Eldar, Aug 20 2020
a(n)/n^2 > 8/Pi^2 for odd n. - M. F. Hasler, Jul 08 2025

A064840 a(n) = tau(n)*sigma(n).

Original entry on oeis.org

1, 6, 8, 21, 12, 48, 16, 60, 39, 72, 24, 168, 28, 96, 96, 155, 36, 234, 40, 252, 128, 144, 48, 480, 93, 168, 160, 336, 60, 576, 64, 378, 192, 216, 192, 819, 76, 240, 224, 720, 84, 768, 88, 504, 468, 288, 96, 1240, 171, 558, 288, 588, 108, 960, 288, 960, 320, 360

Offset: 1

Vladeta Jovovic, Oct 25 2001

Dirichlet convolution of A034761 with (the Dirichlet inverse of A037213). - R. J. Mathar, Feb 11 2011

			For n = 10, a(10) = sigma(10) * tau(10) = 18 * 4 = 72. - _Indranil Ghosh_, Jan 20 2017

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A000005, A000203, A007503, A062354, A062355.

[ NumberOfDivisors(n)*SumOfDivisors(n) : n in [1..40]];

with(numtheory): seq(sigma(n)*tau(n), n=1..58) ; # Zerinvary Lajos, Jun 04 2008

Table[ DivisorSigma[0, n] * DivisorSigma[1, n], {n, 1, 58}] (* Jean-François Alcover, Mar 26 2013 *)

{ for (n=1, 1000, a=numdiv(n)*sigma(n); write("b064840.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

Multiplicative with a(p^e) = (p^(e+1)-1)*(e+1)/(p-1). a(n) = (1/2)*Sum_{i|n, j|n} (i+j).
Dirichlet g.f. (zeta(s)*zeta(s-1))^2/zeta(2s-1). - R. J. Mathar, Feb 11 2011
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (144*Zeta(3)) * (2*log(n) - 1 + 4*gamma - 4*Zeta'(3)/Zeta(3) + 24*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019

A062949 Multiplicative with a(p^e) = ((e+1)p^(e+1)-(e+2)p^e+1)/(p-1).

Original entry on oeis.org

1, 3, 5, 9, 9, 15, 13, 25, 23, 27, 21, 45, 25, 39, 45, 65, 33, 69, 37, 81, 65, 63, 45, 125, 69, 75, 95, 117, 57, 135, 61, 161, 105, 99, 117, 207, 73, 111, 125, 225, 81, 195, 85, 189, 207, 135, 93, 325, 139, 207, 165, 225, 105, 285, 189, 325, 185, 171, 117, 405

Offset: 1

Vladeta Jovovic, Jul 21 2001

Inverse Mobius transform of A062355.

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

Cf. A057660, A029939.

A062949 := proc(n) add(numtheory[phi](d)*numtheory[tau](d), d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Feb 09 2011

f[p_, e_] := ((e+1)*p^(e+1)-(e+2)*p^e+1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 60] (* Amiram Eldar, Jul 31 2019 *)

a(n) = Sum_{d|n} phi(d)*tau(d).
a(n) = Sum_{k=1..n} tau(n/gcd(n, k)).
a(n) = Sum_{d|n} d*uphi(n/d), where uphi() = A047994(). - Vladeta Jovovic, Mar 16 2004

A304408 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*(k_j + 1)).

Original entry on oeis.org

1, 2, 4, 3, 8, 8, 12, 4, 6, 16, 20, 12, 24, 24, 32, 5, 32, 12, 36, 24, 48, 40, 44, 16, 12, 48, 8, 36, 56, 64, 60, 6, 80, 64, 96, 18, 72, 72, 96, 32, 80, 96, 84, 60, 48, 88, 92, 20, 18, 24, 128, 72, 104, 16, 160, 48, 144, 112, 116, 96, 120, 120, 72, 7, 192, 160, 132, 96, 176, 192

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(20) = a(2^2*5) = (2 - 1)*(2 + 1) * (5 - 1)*(1 + 1) = 24.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000010, A000026, A000040, A000302 (numbers n such that a(n) is odd), A001221, A006093, A007947, A023900, A034444, A059975, A062355, A173557, A304407, A304409, A304411, A304412.

a:= n-> mul((i[1]-1)*(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021

a[n_] := Times @@ ((#[[1]] - 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 70}]
Table[DivisorSigma[0, n] EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]], {n, 70}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*(e+1))} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*abs(A023900(n)) = A000005(n)*A173557(n) = A000005(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*phi(n) if n is a squarefree (A005117), where omega() = A001221 and phi() = A000010.

A327169 Number of distinct k such that A000005(k)*A000010(k) is equal to n.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Sep 19 2019

nonn

a(n) tells how many times in total n occurs in A062355.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000005, A000010, A062355.

Cf. also A327153, A327166.

A327169(n) = sum(k=1,n,(eulerphi(k)*numdiv(k))==n);

f(d, m) = my(v = invphi(d)); sum(i = 1, #v, numdiv(v[i]) == m); \\ using Max Alekseyev's invphi.gp
a(n) = sumdiv(n, d, f(d, n/d)); \\ Amiram Eldar, Feb 01 2025

a(n) = Sum_{k=1..n} [A000005(k)*A000010(k) == n], where [ ] is the Iverson bracket.

A335707 Decimal expansion of Sum_{primes p} log(p) / (p^2 + p - 1).

Original entry on oeis.org

4, 1, 8, 7, 5, 7, 5, 7, 8, 7, 9, 4, 1, 2, 5, 4, 8, 0, 5, 3, 4, 4, 2, 1, 2, 5, 6, 0, 2, 8, 7, 0, 4, 6, 3, 6, 1, 3, 6, 5, 5, 5, 1, 6, 5, 4, 4, 9, 2, 8, 7, 0, 2, 9, 4, 0, 5, 2, 2, 0, 0, 2, 8, 0, 3, 7, 7, 5, 4, 9, 6, 9, 2, 5, 9, 5, 2, 8, 9, 0, 8, 0, 2, 1, 4, 8, 0, 6, 7, 2, 8, 4, 7, 7, 8, 5, 1, 1, 8, 8, 8, 5, 9, 4, 0, 0

Offset: 0

Vaclav Kotesovec, Jun 18 2020

			0.41875757879412548053442125602870463613655516544928702940522...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 100, p. 169.

Cf. A062355, A085609.

ratfun = 1 / (p^2 + p - 1); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 100]], {m, 2000, 20000, 2000}]

A079535 a(n) = phi(n)*d(n) - n.

Original entry on oeis.org

0, 0, 1, 2, 3, 2, 5, 8, 9, 6, 9, 12, 11, 10, 17, 24, 15, 18, 17, 28, 27, 18, 21, 40, 35, 22, 45, 44, 27, 34, 29, 64, 47, 30, 61, 72, 35, 34, 57, 88, 39, 54, 41, 76, 99, 42, 45, 112, 77, 70, 77, 92, 51, 90, 105, 136, 87, 54, 57, 132, 59, 58, 153, 160, 127, 94, 65, 124, 107, 122, 69

Offset: 1

N. J. A. Sloane, Jan 23 2003

nonn

It is known that a(n) >= 0.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 10.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000005, A062355.

[EulerPhi(n)*DivisorSigma(0,n) - n: n in [1..80]]; // G. C. Greubel, Jan 14 2019

Table[EulerPhi[n]*DivisorSigma[0, n] - n, {n, 1, 80}] (* G. C. Greubel, Jan 14 2019 *)

vector(80, n, eulerphi(n)*sigma(n,0) - n) \\ G. C. Greubel, Jan 14 2019

[euler_phi(n)*sigma(n, 0) - n for n in (1..80)] # G. C. Greubel, Jan 14 2019

A079536 a(n) = phi(n)*d(n) - sigma(n).

Original entry on oeis.org

0, -1, 0, -1, 2, -4, 4, 1, 5, -2, 8, -4, 10, 0, 8, 9, 14, -3, 16, 6, 16, 4, 20, 4, 29, 6, 32, 16, 26, -8, 28, 33, 32, 10, 48, 17, 34, 12, 40, 38, 38, 0, 40, 36, 66, 16, 44, 36, 69, 27, 56, 46, 50, 24, 88, 72, 64, 22, 56, 24, 58, 24, 112, 97, 108, 16, 64, 66, 80, 48, 68, 93, 70, 30, 116, 76, 144

Offset: 1

N. J. A. Sloane, Jan 23 2003

sign

It is known that a(n) >= 0 if n is odd

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 10.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000005, A000203, A062355, A079535, A079537.

[EulerPhi(n)*DivisorSigma(0,n) - DivisorSigma(1,n): n in [1..80]]; // G. C. Greubel, Jan 14 2019

Table[EulerPhi[n]*DivisorSigma[0, n] - DivisorSigma[1, n], {n,1,80}] (* G. C. Greubel, Jan 14 2019 *)

vector(80, n, eulerphi(n)*sigma(n,0) - sigma(n,1)) \\ G. C. Greubel, Jan 14 2019

[euler_phi(n)*sigma(n, 0) - sigma(n,1) for n in (1..80)] # G. C. Greubel, Jan 14 2019

A318893 Filter sequence combining the prime signature of n (A046523) with Euler totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 21, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 34, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 42, 48, 43, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 53, 70, 59, 71, 66, 72, 73, 74, 51, 75, 76, 77, 78, 79, 80, 81, 76, 82, 83, 71

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Restricted growth sequence transform of A286160.

For all i, j: a(i) = a(j) => A062355(i) = A062355(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A046523, A286160, A318839, A318892.

Cf. also A061468, A062355.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A318893aux(n) = [eulerphi(n), A046523(n)];
v318893 = rgs_transform(vector(up_to,n,A318893aux(n)));
A318893(n) = v318893[n];

A327171 a(n) = phi(n) * core(n), where phi is Euler totient function, and core gives the squarefree part of n.

Original entry on oeis.org

1, 2, 6, 2, 20, 12, 42, 8, 6, 40, 110, 12, 156, 84, 120, 8, 272, 12, 342, 40, 252, 220, 506, 48, 20, 312, 54, 84, 812, 240, 930, 32, 660, 544, 840, 12, 1332, 684, 936, 160, 1640, 504, 1806, 220, 120, 1012, 2162, 48, 42, 40, 1632, 312, 2756, 108, 2200, 336, 2052, 1624, 3422, 240, 3660, 1860, 252, 32, 3120, 1320

Offset: 1

Antti Karttunen, Sep 28 2019

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 161.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Chantal David and Francesco Pappalardi, Average Frobenius distributions of elliptic curves, International Mathematics Research Notices, Vol. 1999, No. 4 (1999), pp. 165-183, alternative link.

Cf. A082473 (gives the terms in ascending order, with duplicates removed).
Cf. A000010, A007913, A053143, A248003, A327170, A327172.
Cf. also A002618, A062355.

[EulerPhi(n)*Squarefree(n): n in [1..100]]; // G. C. Greubel, Jul 13 2024

Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, 66] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)

PARI
```
A327171(n) = eulerphi(n)*core(n);
```

A327171(n) = { my(f=factor(n)); prod (i=1, #f~, (f[i, 1]-1)*(f[i, 1]^(-1 + f[i, 2] + (f[i, 2]%2)))); };

from sympy.ntheory.factor_ import totient, core
def A327171(n):
    return totient(n)*core(n) # Chai Wah Wu, Sep 29 2019

[euler_phi(n)*squarefree_part(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024

a(n) = A000010(n) * A007913(n).
Multiplicative with a(p^k) = (p-1) * p^((k-1)+(k mod 2)).
Sum_{n>=1} 1/a(n) = (Pi^2/6) * Product_{p prime} (1 + (p+1)/(p^2*(p-1))) = 3.96555686901754604330... - Amiram Eldar, Oct 16 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.1500809164... . - Amiram Eldar, Dec 05 2022
a(n) = A000010(A053143(n)). - Amiram Eldar, Sep 15 2023

cp's OEIS Frontend

A062354 a(n) = sigma(n)*phi(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A064840 a(n) = tau(n)*sigma(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A062949 Multiplicative with a(p^e) = ((e+1)*p^(e+1)-(e+2)*p^e+1)/(p-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A304408 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*(k_j + 1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A327169 Number of distinct k such that A000005(k)*A000010(k) is equal to n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A335707 Decimal expansion of Sum_{primes p} log(p) / (p^2 + p - 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A079535 a(n) = phi(n)*d(n) - n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

A062949 Multiplicative with a(p^e) = ((e+1)p^(e+1)-(e+2)p^e+1)/(p-1).