A319087 -id:A319087 - cp's OEIS frontend

A053191 a(n) = n^2 * phi(n).

1, 4, 18, 32, 100, 72, 294, 256, 486, 400, 1210, 576, 2028, 1176, 1800, 2048, 4624, 1944, 6498, 3200, 5292, 4840, 11638, 4608, 12500, 8112, 13122, 9408, 23548, 7200, 28830, 16384, 21780, 18496, 29400, 15552, 49284, 25992, 36504, 25600, 67240

Offset: 1

Labos Elemer, Mar 02 2000

Number of invertible 2 X 2 symmetric matrices over Z(n). - T. D. Noe, Jan 13 2006
Note that A115077 gives the number of 2 X 2 symmetric matrices having nonzero determinant. However, for composite n, a nonzero determinant is not sufficient for the matrix to be invertible; the determinant must also be relatively prime to n. - T. D. Noe, Jan 13 2006
Also Euler phi function of n^3.
For n^k, EulerPhi(n^k) = n^(k-1)*EulerPhi(n). The same holds if Phi is replaced by the cototient function.
Also, the sum of the degrees of the irreducible representations of the group GL(2,Z_n) (sequence A000252). - Sharon Sela (sharonsela(AT)hotmail.com), Feb 06 2002

			n=5: n^3 = 125, EulerPhi(125) = 125 - 25 = 100.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000252 (number of invertible 2 X 2 matrices over Z(n)), A115075, A115076, A115077.

Cf. A000010, A051953, A002618, A053650, A053192, A001248, A319087.

[ n^2*EulerPhi(n) : n in [1..100] ]; // Vincenzo Librandi, Apr 21 2011

with(numtheory):a:=n->phi(n^3): seq(a(n), n=1..41); # Zerinvary Lajos, Oct 07 2007

Table[cnt=0; Do[m={{a, b}, {b, c}}; If[Det[m, Modulus->n]>0 && MatrixQ[Inverse[m, Modulus->n]], cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}]; cnt, {n, 2, 50}] (* T. D. Noe, Jan 13 2006 *)
Table[n^2*EulerPhi[n],{n,1,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

a(n) = n^2*eulerphi(n); \\ Michel Marcus, Oct 31 2017

[n^2*euler_phi(n) for n in range(1, 42)] # Zerinvary Lajos, Jun 06 2009

a(n) = n^2 * phi(n) = A000010(n^3).
Dirichlet g.f.: zeta(s-3)/zeta(s-2). - R. J. Mathar, Feb 09 2011
The n-th term of the Dirichlet inverse is n^2 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221. - Álvar Ibeas, Nov 24 2017
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^4 - p^3 - p + 1)) = 1.38097852211302096879... - Amiram Eldar, Dec 06 2020

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007

A011755 a(n) = Sum_{k=1..n} k*phi(k).

Original entry on oeis.org

1, 3, 9, 17, 37, 49, 91, 123, 177, 217, 327, 375, 531, 615, 735, 863, 1135, 1243, 1585, 1745, 1997, 2217, 2723, 2915, 3415, 3727, 4213, 4549, 5361, 5601, 6531, 7043, 7703, 8247, 9087, 9519, 10851, 11535, 12471, 13111, 14751, 15255, 17061, 17941, 19021, 20033

Offset: 1

N. J. A. Sloane

nonn

a(n) = Sum_{(x,y): 1<=x<=y<=n, 1=gcd(x,y)} y. Sum_{(x,y): 1<=x<=y<=n, 1=gcd(x,y)} x = (a(n)+1)/2. - Vladeta Jovovic, Jan 02 2003

Equals row sums of triangle A110663. Example: a(4) = 17 = (6 + 5 + 4 + 2), where row 4 of triangle A110663 = (6, 5, 4, 2). - Gary W. Adamson, Jul 26 2008

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..2001 from Indranil Ghosh)

Partial sums of A002618.

Cf. A000010, A002088, A110663, A319087.

Accumulate[Table[k*EulerPhi[k], {k, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)

a(n) = sum(k=1, n, k*eulerphi(k)); \\ Michel Marcus, Feb 13 2017

from sympy import totient
def A011755(n): return sum(k*totient(k) for k in range(1,n+1)) # Chai Wah Wu, Feb 21 2023

Asymptotically: a(n) ~ C*n^3 with C=0.20264.... - Benoit Cloitre, Jan 14 2002
Asymptotically: a(n) ~ (2/Pi^2)*n^3. - Vladeta Jovovic, Jan 02 2003
a(n) = Sum_{k=1..n} phi(k^2). - Vaclav Kotesovec, May 08 2024

A333292 Triangle read by rows: T(m,n) = Sum_{ 1 <= i <= m, 1 <= j <= n, gcd(i,j)=1 } i*j, for 1 <= n <= m.

Original entry on oeis.org

1, 3, 5, 6, 14, 23, 10, 18, 39, 55, 15, 33, 69, 105, 155, 21, 39, 75, 111, 191, 227, 28, 60, 117, 181, 296, 374, 521, 36, 68, 149, 213, 368, 446, 649, 777, 45, 95, 176, 276, 476, 554, 820, 1020, 1263, 55, 105, 216, 316, 516, 594, 930, 1130, 1463, 1663, 66, 138, 282, 426, 681, 825, 1238, 1526, 1958, 2268, 2873

Offset: 1

N. J. A. Sloane, Mar 23 2020

The last two diagonals are A333293, Sum_{k=1..n-1} k^2*phi(k) + n^2*phi(n)/2, and A319087, Sum_{k=1..n} k^2*phi(k), where phi = A000010. Is there a similar formula for the general term?

			Triangle begins:
1,
3, 5,
6, 14, 23,
10, 18, 39, 55,
15, 33, 69, 105, 155,
21, 39, 75, 111, 191, 227,
28, 60, 117, 181, 296, 374, 521,
36, 68, 149, 213, 368, 446, 649, 777,
45, 95, 176, 276, 476, 554, 820, 1020, 1263,
55, 105, 216, 316, 516, 594, 930, 1130, 1463, 1663,
...

Alois P. Heinz, Rows n = 1..141, flattened

First two columns are A000217 and A074378, rightmost two diagonals are A333293 and A319087.
Main diagonal is A319087.
Cf. A320541.

T:= (m, n)-> add(add(`if`(igcd(i, j)=1, i*j, 0), j=1..n), i=1..m):
seq(seq(T(m, n), n=1..m), m=1..12);  # Alois P. Heinz, Mar 23 2020

A333293 a(n) = Sum_{k=1..n-1} k^2phi(k) + n^2phi(n)/2, where phi = A000010.

Original entry on oeis.org

3, 14, 39, 105, 191, 374, 649, 1020, 1463, 2268, 3161, 4463, 6065, 7553, 9477, 12813, 16097, 20318, 25167, 29413, 34479, 42718, 50841, 59395, 69701, 80318, 91583, 108061, 123435, 141450, 164057, 183139, 203277, 227225, 249701, 282119, 319757, 351005, 382057, 428477, 472681, 522094, 580283, 623943, 671519

Offset: 2

N. J. A. Sloane, Mar 23 2020

nonn

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A319087, A000010.

Next-to-last diagonal of A333292.

P:= [seq(k^2*numtheory:-phi(k),k=1..100)]:
T:= ListTools:-PartialSums(P):
seq(T[i-1]+P[i]/2,i=2..100); # Robert Israel, Mar 24 2020

a(n) = sum(k=1, n-1, k^2*eulerphi(k)) + n^2*eulerphi(n)/2; \\ Michel Marcus, Mar 23 2020

A333291 a(n) = Sum_{i=1..n, gcd(i,n)=1} i*phi(i) where phi is Euler's totient function A000010.

Original entry on oeis.org

1, 1, 3, 7, 17, 21, 49, 69, 105, 103, 217, 173, 375, 347, 435, 509, 863, 601, 1243, 983, 1271, 1265, 2217, 1449, 2575, 2225, 2935, 2573, 4549, 2241, 5601, 4609, 5195, 4997, 6453, 4531, 9519, 7099, 8457, 6897, 13111, 6621, 15255, 11053, 11691, 12397, 20033, 11471, 20905, 14563, 19307, 17663, 28901, 16285, 26119

Offset: 1

N. J. A. Sloane, Mar 22 2020

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Suggested by A023896 and A319087.

Cf. A000010, A011755.

a:= n-> add(`if`(igcd(i, n)=1, i*numtheory[phi](i), 0), i=1..n):
seq(a(n), n=1..55);  # Alois P. Heinz, Mar 22 2020

a[n_] := Sum[If[CoprimeQ[i, n], i * EulerPhi[i], 0], {i, 1, n}]; Array[a, 100] (* Amiram Eldar, Dec 01 2024 *)

a(n) = sum(i=1, n, if (gcd(n, i) == 1, i*eulerphi(i))); \\ Michel Marcus, Mar 23 2020

A333297 a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} i.

Original entry on oeis.org

1, 4, 13, 25, 55, 73, 136, 184, 265, 325, 490, 562, 796, 922, 1102, 1294, 1702, 1864, 2377, 2617, 2995, 3325, 4084, 4372, 5122, 5590, 6319, 6823, 8041, 8401, 9796, 10564, 11554, 12370, 13630, 14278, 16276, 17302, 18706, 19666, 22126, 22882, 25591, 26911, 28531, 30049, 33292, 34444, 37531, 39031

Offset: 1

N. J. A. Sloane, Mar 25 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000010, A002618, A023896, A115004, A018805, A319087, A333295.

Vi := proc(m,n) local a,i,j; a:=0;
for i from 1 to m do for j from 1 to n do
if igcd(i,j)=1 then a:=a+i; fi; od: od: a; end;
# the diagonal :
[seq(Vi(n,n),n=1..50)];
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n,
      a(n-1) + 3*n*numtheory[phi](n)/2)
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Mar 25 2020

a[n_] := a[n] = If[n < 2, n, a[n - 1] + 3 n EulerPhi[n]/2];
Array[a, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

a(n)={my(s=0);for(i=1,n,for(j=1,n,if(gcd(i,j)==1,s+=i)));s};
for(k=1,45,print1(a(k),", ")) \\ Hugo Pfoertner, Mar 25 2020

From Alois P. Heinz, Mar 25 2020: (Start)
a(n) = a(n-1) + 3*n*phi(n)/2 for n > 1, a(n) = n for n <= 1.
a(n) = 1 + Sum_{k=2..n} 3*k*phi(k)/2. (End)
a(n) = a(n-1) + 3 * A023896(n) for n > 1. - Hugo Pfoertner, Mar 26 2020
a(n) ~ (3/Pi^2) * n^3. - Amiram Eldar, Dec 01 2024

A344526 a(n) = Sum_{k=1..n} k^3 * phi(k).

Original entry on oeis.org

1, 9, 63, 191, 691, 1123, 3181, 5229, 9603, 13603, 26913, 33825, 60189, 76653, 103653, 136421, 215029, 250021, 373483, 437483, 548615, 655095, 922769, 1033361, 1345861, 1556773, 1911067, 2174491, 2857383, 3073383, 3967113, 4491401, 5210141, 5839005, 6868005, 7427877, 9251385

Offset: 1

Seiichi Manyama, May 22 2021

nonn

Cf. A000010, A011755, A068970, A319087, A344522.

a[n_] := Sum[k^3 * EulerPhi[k], {k, 1, n}]; Array[a, 40] (* Amiram Eldar, May 22 2021 *)
Accumulate[Table[k^3*EulerPhi[k], {k, 1, 40}]] (* Vaclav Kotesovec, May 22 2021 *)

PARI
```
a(n) = sum(k=1, n, k^3*eulerphi(k));
```

a(n) ~ 6*n^5 / (5*Pi^2). - Vaclav Kotesovec, May 22 2021

cp's OEIS Frontend

A053191 a(n) = n^2 * phi(n).

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A011755 a(n) = Sum_{k=1..n} k*phi(k).

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A333292 Triangle read by rows: T(m,n) = Sum_{ 1 <= i <= m, 1 <= j <= n, gcd(i,j)=1 } i*j, for 1 <= n <= m.

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A333293 a(n) = Sum_{k=1..n-1} k^2*phi(k) + n^2*phi(n)/2, where phi = A000010.

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A333291 a(n) = Sum_{i=1..n, gcd(i,n)=1} i*phi(i) where phi is Euler's totient function A000010.

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A333297 a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} i.

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A344526 a(n) = Sum_{k=1..n} k^3 * phi(k).

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A333293 a(n) = Sum_{k=1..n-1} k^2phi(k) + n^2phi(n)/2, where phi = A000010.