A062949 -id:A062949 - cp's OEIS frontend

A127469 a(n) = n * A062949(n).

1, 6, 15, 36, 45, 90, 91, 200, 207, 270, 231, 540, 325, 546, 675, 1040, 561, 1242, 703, 1620, 1365, 1386, 1035, 3000, 1725, 1950, 2565, 3276, 1653, 4050, 1891, 5152, 3465, 3366, 4095, 7452, 2701, 4218, 4875, 9000, 3321, 8190, 3655, 8316, 9315, 6210, 4371, 15600

Offset: 1

Gary W. Adamson, Jan 15 2007

			a(5) = 45 = 5 * A062949(5) = 5 * 9.

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

Cf. A062949.

Row sums of triangle A127470.

A127469 := proc(n) n*A062949(n) ; end proc:
seq(A127469(n),n=1..80) ; # 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_] := n * Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 31 2023 *)

a(n) = n * A062949(n).

Multiplicative with a(p^e) = ((e+1)*p^(2*e+1) - (e+2)*p^(2*e) + p^e)/(p-1). - Amiram Eldar, Aug 31 2023

More terms from Amiram Eldar, Aug 31 2023

A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.

Original entry on oeis.org

1, 2, 4, 6, 8, 8, 12, 16, 18, 16, 20, 24, 24, 24, 32, 40, 32, 36, 36, 48, 48, 40, 44, 64, 60, 48, 72, 72, 56, 64, 60, 96, 80, 64, 96, 108, 72, 72, 96, 128, 80, 96, 84, 120, 144, 88, 92, 160, 126, 120, 128, 144, 104, 144, 160, 192, 144, 112, 116, 192, 120, 120, 216, 224

Offset: 1

Jason Earls, Jul 06 2001

a(n) = sum of gcd(k-1,n) for 1 <= k <= n and gcd(k,n)=1 (Menon's identity).
For n = 2^(4*k^2 - 1), k >= 1, the terms of the sequence are square and for n = 2^((3*k + 2)^3 - 1), k >= 1, the terms of the sequence are cubes. - Marius A. Burtea, Nov 14 2019
Sum_{k>=1} 1/a(k) diverges. - Vaclav Kotesovec, Sep 20 2020

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.
P. K. Menon, On the sum gcd(a-1,n) [(a,n)=1], J. Indian Math. Soc. (N.S.), 29 (1965), 155-163.
József Sándor, On Dedekind's arithmetical function, Seminarul de teoria structurilor (in Romanian), No. 51, Univ. Timișoara, 1988, pp. 1-15. See p. 11.
József Sándor, Some diophantine equations for particular arithmetic functions (in Romanian), Seminarul de teoria structurilor, No. 53, Univ. Timișoara, 1989, pp. 1-10. See p. 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith)
Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019.
Vaclav Kotesovec, Graph - the asymptotic ratio (250000000 terms)
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, Section 3.15.
R. Sivaramakrishnan, Problem E 1962, Elementary Problems, The American Mathematical Monthly, Vol. 74, No. 2 (1967), p. 198; Solution, ibid., Vol. 75, No. 5 (1968), p. 550.
Marius Tarnauceanu, A generalization of the Menon's identity, arXiv:1109.2198 [math.GR], 2011-2012.
Laszlo Toth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
Wikipedia, Arithmetic function (Menon's identity).

Cf. A003557, A173557, A061468, A062816, A079535, A062949 (inverse Mobius transform), A304408, A318519, A327169 (number of times n occurs in this sequence).

Cf. A062354, A064840.

[NumberOfDivisors(n)*EulerPhi(n):n in [1..65]]; // Marius A. Burtea, Nov 14 2019

seq(tau(n)*phi(n), n=1..64); # Zerinvary Lajos, Jan 22 2007

Table[EulerPhi[n] DivisorSigma[0, n], {n, 80}] (* Carl Najafi, Aug 16 2011 *)
f[p_, e_] := (e+1)*(p-1)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

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

{ for (n=1, 1000, write("b062355.txt", n, " ", numdiv(n)*eulerphi(n)) ) } \\ Harry J. Smith, Aug 05 2009

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X^2)/(1 - p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 15 2020

Dirichlet convolution of A047994 and A000010. - R. J. Mathar, Apr 15 2011
a(n) = A000005(n)*A000010(n). Multiplicative with a(p^e) = (e+1)*(p-1)*p^(e-1). - R. J. Mathar, Jun 23 2018
a(n) = A173557(n) * A318519(n) = A003557(n) * A304408(n). - Antti Karttunen, Sep 16 2018 & Sep 20 2019
From Vaclav Kotesovec, Jun 15 2020: (Start)
Let f(s) = Product_{primes p} (1 - 2*p^(-s) + p^(1-2*s)).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ n^2 * (f(2)*(log(n)/2 + gamma - 1/4) + f'(2)/2), where f(2) = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.42824950567709444...,
f'(2) = 2 * A065464 * A335707 = f(2) * Sum_{primes p} 2*log(p) / (p^2 + p - 1) = 0.35866545223424232469545420783620795... and gamma is the Euler-Mascheroni constant A001620. (End)
From Amiram Eldar, Mar 02 2021: (Start)
a(n) >= n (Sivaramakrishnan, 1967).
a(n) >= sigma(n), for odd n (Sándor, 1988).
a(n) >= phi(n) + n - 1 (Sándor, 1989) (End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} uphi(gcd(n,k)), where uphi(n) = A047994(n).
a(n) = Sum_{k=1..n} uphi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A062380 a(n) = Sum_{i|n,j|n} phi(i)*phi(j)/phi(gcd(i,j)), where phi is Euler totient function.

Original entry on oeis.org

1, 4, 7, 14, 13, 28, 19, 42, 37, 52, 31, 98, 37, 76, 91, 114, 49, 148, 55, 182, 133, 124, 67, 294, 113, 148, 163, 266, 85, 364, 91, 290, 217, 196, 247, 518, 109, 220, 259, 546, 121, 532, 127, 434, 481, 268, 139, 798, 229, 452, 343, 518, 157, 652, 403, 798, 385

Offset: 1

Vladeta Jovovic, Jul 07 2001

A176003 is a subsequence. - Peter Luschny, Sep 12 2012

			Let p be a prime then a(p) = phi(1)*tau(1)+phi(p)*tau(p^2) = 1+(p-1)*3 = 3*p-2. - _Peter Luschny_, Sep 12 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A060648, A062949.

with(numtheory):
a:= n->  add(phi(d)*tau(d^2), d=divisors(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 12 2012

a[n_] := DivisorSum[n, EulerPhi[#] DivisorSigma[0, #^2]&]; Array[a, 60] (* Jean-François Alcover, Dec 05 2015 *)
f[p_, e_] := ((2*e+1)*p^(e+1) - (2*e+3)*p^e + 2)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n)=sumdiv(n,i,eulerphi(i)*sumdiv(n,j,eulerphi(j)/eulerphi(gcd(i,j)))) \\ Charles R Greathouse IV, Sep 12 2012

def A062380(n) :
    d = divisors(n); cp = cartesian_product([d, d])
    return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))
[A062380(n) for n in (1..57)]  # Peter Luschny, Sep 10 2012

a(n) = Sum_{d|n} phi(d)*tau(d^2).
Multiplicative with a(p^e) = 1 + Sum_{k=1..e} (2k+1)(p^k-p^{k-1}) = ((2e+1)p^(e+1) - (2e+3)p^e+2)/(p-1). - Mitch Harris, May 24 2005
a(n) = Sum_{c|n,d|n} phi(lcm(c,d)). - Peter Luschny, Sep 10 2012
a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^2 ). - Seiichi Manyama, May 19 2024

A372997 a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^3 ).

Original entry on oeis.org

1, 5, 9, 19, 17, 45, 25, 59, 51, 85, 41, 171, 49, 125, 153, 163, 65, 255, 73, 323, 225, 205, 89, 531, 157, 245, 231, 475, 113, 765, 121, 419, 369, 325, 425, 969, 145, 365, 441, 1003, 161, 1125, 169, 779, 867, 445, 185, 1467, 319, 785, 585, 931, 209, 1155, 697, 1475

Offset: 1

Seiichi Manyama, May 19 2024

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

Cf. A062380, A062949, A372999.

Cf. A000005, A000010, A344221.

f[p_, e_] := (3 - (3*e+4)*p^e + (3*e+1)*p^(e+1))/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d^3));

If p is prime, a(p) = 4*p - 3.
a(n) = Sum_{d|n} phi(d) * tau(d^3).
Multiplicative with a(p^e) = (3 - (3*e+4)*p^e + (3*e+1)*p^(e+1))/(p-1). - Amiram Eldar, May 21 2024

A372999 a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^4 ).

Original entry on oeis.org

1, 6, 11, 24, 21, 66, 31, 76, 65, 126, 51, 264, 61, 186, 231, 212, 81, 390, 91, 504, 341, 306, 111, 836, 201, 366, 299, 744, 141, 1386, 151, 548, 561, 486, 651, 1560, 181, 546, 671, 1596, 201, 2046, 211, 1224, 1365, 666, 231, 2332, 409, 1206, 891, 1464, 261, 1794

Offset: 1

Seiichi Manyama, May 19 2024

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

Cf. A062380, A062949, A372997.

f[p_, e_] := (4 - (4*e+5)*p^e + (4*e+1)*p^(e+1))/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d^4));

If p is prime, a(p) = 5*p - 4.
a(n) = Sum_{d|n} phi(d) * tau(d^4).
Multiplicative with a(p^e) = (4 - (4*e+5)*p^e + (4*e+1)*p^(e+1))/(p-1). - Amiram Eldar, May 21 2024

A127472 Triangle T(n,k) = Sum_{j=k..n, j|n, k|j} phi(j) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 3, 0, 2, 4, 3, 0, 2, 5, 0, 0, 0, 4, 6, 3, 4, 0, 0, 2, 7, 0, 0, 0, 0, 0, 6, 8, 7, 0, 6, 0, 0, 0, 4, 9, 0, 8, 0, 0, 0, 0, 0, 6, 10, 5, 0, 0, 8, 0, 0, 0, 0, 4, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 12, 9, 8, 6, 0, 6, 0, 0, 0, 0, 0, 4, 13

Offset: 1

Gary W. Adamson, Jan 15 2007

Defined by the matrix product A054522 * A051731.

			First few rows of the triangle are;
.1;
.2, 1;
.3, 0, 2;
.4, 3, 0, 2;
.5, 0, 0, 0, 4;
.6, 3, 4, 0, 0, 2;
.7, 0, 0, 0, 0, 0, 6;
.8, 7, 0, 6, 0, 0, 0, 4;
....

Cf. A054522, A051731, A062949 (row sums), A000010 (diagonal n=k), A127471 (swapped matrix product).

A127472 := proc(n,k)
        a := 0 ;
        for j from k to n do
                if (n mod j = 0 ) and (j mod k =0 ) then
                        a := a+numtheory[phi](j) ;
                end if;
        end do;
        a ;
end proc:
seq(seq(A127472(n,k),k=1..n),n=1..14) ; # R. J. Mathar, Nov 11 2011

T(n,k) = Sum_{j=k..n} A054522(n,j) * A051731(j,k), 1<=k<=n.

A333645 a(n) = Sum_{d|n} uphi(d).

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 12, 11, 10, 11, 15, 13, 14, 15, 27, 17, 22, 19, 25, 21, 22, 23, 36, 29, 26, 37, 35, 29, 30, 31, 58, 33, 34, 35, 55, 37, 38, 39, 60, 41, 42, 43, 55, 55, 46, 47, 81, 55, 58, 51, 65, 53, 74, 55, 84, 57, 58, 59, 75, 61, 62, 77, 121, 65, 66, 67, 85, 69, 70

Offset: 1

Ilya Gutkovskiy, Mar 31 2020

Inverse Moebius transform of A047994.

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

Cf. A005117 (fixed points), A023900, A047994, A055653, A062949, A065464, A152649.

uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[uphi[d], {d, Divisors[n]}]; Table[a[n], {n, 70}]
A023900[n_] := Sum[MoebiusMu[d] d, {d, Divisors[n]}]; A062949[n_] := Sum[EulerPhi[d] DivisorSigma[0, d], {d, Divisors[n]}]; a[n_] := Sum[A023900[d] A062949[n/d], {d, Divisors[n]}]; Table[a[n], {n, 70}]
f[p_,e_] := (p^(e+1) - e*p + e - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100] (* Amiram Eldar, Nov 12 2022 *)

uphi(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); \\ A047994
a(n) = sumdiv(n, d, uphi(d)); \\ Michel Marcus, Mar 31 2020

G.f.: Sum_{k>=1} uphi(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A023900(d) * A062949(n/d).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(e+1) - e*p + e - 1)/(p-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - (2*p-1)/p^3) = A152649 * A065464 = 0.5793804872... . (End)

A341636 a(n) = Sum_{d|n} phi(d) * tau(d) * tau(n/d).

Original entry on oeis.org

1, 4, 6, 13, 10, 24, 14, 38, 29, 40, 22, 78, 26, 56, 60, 103, 34, 116, 38, 130, 84, 88, 46, 228, 79, 104, 124, 182, 58, 240, 62, 264, 132, 136, 140, 377, 74, 152, 156, 380, 82, 336, 86, 286, 290, 184, 94, 618, 153, 316, 204, 338, 106, 496, 220, 532, 228, 232, 118, 780, 122, 248

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Inverse Moebius transform of A062949.

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

Cf. A000005, A000010, A000203, A007426, A055507, A062355, A062949.

Table[Sum[EulerPhi[d] DivisorSigma[0, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 62}]
Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[0, n/GCD[n, k]], {k, n}], {n, 62}]
f[p_, e_] := (p + 1 + e*(p - 1) + p^(e + 1)*(e*(p - 1) + p - 3))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * tau(n/gcd(n,k)).
a(n) = Sum_{d|n} A062949(d).
Multiplicative with a(p^e) = (p + 1 + e*(p-1) + p^(e+1)*(e*(p-1)+p-3))/(p-1)^2. - Amiram Eldar, Sep 15 2023

A372226 a(n) = Sum_{k=1..n} sigma_2( n/gcd(k,n) ).

Original entry on oeis.org

1, 6, 21, 48, 105, 126, 301, 388, 567, 630, 1221, 1008, 2041, 1806, 2205, 3116, 4641, 3402, 6517, 5040, 6321, 7326, 11661, 8148, 13125, 12246, 15327, 14448, 23577, 13230, 28861, 24956, 25641, 27846, 31605, 27216, 49321, 39102, 42861, 40740, 67281, 37926

Offset: 1

Seiichi Manyama, May 19 2024

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

Cf. A062949, A062952.
Cf. A064987.
Cf. A002117, A013662.

a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[2, #] &]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d, 2));

a(n) = Sum_{d|n} phi(d) * sigma_2(d).
From Amiram Eldar, May 20 2024: (Start)
Multiplicative with a(p^e) = (p^(3*e+5) - p^(3*e+4) - p^(e+3) + p^e + p^4 - p^2) / ((p^2 - 1) * (p^3 - 1)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.749582840863254826301... . (End)

A373002 a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^n ).

Original entry on oeis.org

1, 4, 9, 24, 25, 120, 49, 144, 135, 540, 121, 1728, 169, 1456, 2145, 800, 289, 5220, 361, 8840, 5985, 5544, 529, 21216, 1125, 9100, 1863, 25200, 841, 252000, 961, 4160, 23529, 20196, 31465, 94392, 1369, 28120, 38961, 113520, 1681, 991452, 1849, 101024, 118215

Offset: 1

Seiichi Manyama, May 19 2024

nonn

Cf. A062380, A062949, A372997, A372999.

Cf. A344223, A373003.

a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d^n));

a(n) = n * A373003(n).
If p is prime, a(p) = p^2.
a(n) = Sum_{d|n} phi(d) * tau(d^n).

cp's OEIS Frontend

A127469 a(n) = n * A062949(n).

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A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.

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A062380 a(n) = Sum_{i|n,j|n} phi(i)*phi(j)/phi(gcd(i,j)), where phi is Euler totient function.

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A372997 a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^3 ).

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A372999 a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^4 ).

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A127472 Triangle T(n,k) = Sum_{j=k..n, j|n, k|j} phi(j) read by rows, 1<=k<=n.

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A333645 a(n) = Sum_{d|n} uphi(d).

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