A144613 -id:A144613 - cp's OEIS frontend

A283123 a(n) = sigma(9*n).

13, 39, 40, 91, 78, 120, 104, 195, 121, 234, 156, 280, 182, 312, 240, 403, 234, 363, 260, 546, 320, 468, 312, 600, 403, 546, 364, 728, 390, 720, 416, 819, 480, 702, 624, 847, 494, 780, 560, 1170, 546, 960, 572, 1092, 726, 936, 624, 1240, 741, 1209

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

In general, for k>=1, Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k). - Vaclav Kotesovec, May 11 2024

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

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), this sequence (k=9).

Cf. A008591.

Table[DivisorSigma[1, 9 n], {n, 50}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(9*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(9*n).

Sum_{k=1..n} a(k) = (35*Pi^2/36) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A283118 a(n) = sigma(5*n).

Original entry on oeis.org

6, 18, 24, 42, 31, 72, 48, 90, 78, 93, 72, 168, 84, 144, 124, 186, 108, 234, 120, 217, 192, 216, 144, 360, 156, 252, 240, 336, 180, 372, 192, 378, 288, 324, 248, 546, 228, 360, 336, 465, 252, 576, 264, 504, 403, 432, 288, 744, 342, 468, 432, 588, 324, 720

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

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

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), this sequence (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).

Cf. A008587.

Table[DivisorSigma[1, 5 n], {n, 56}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(5*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(5*n).

Sum_{k=1..n} a(k) = (29*Pi^2/60) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

Original entry on oeis.org

1, -4, 2, 9, -9, -2, 0, -5, 9, 9, 0, -9, -1, -9, 0, -1, 9, 9, -9, 9, 0, 9, -5, -18, -18, 9, 7, 0, 9, 0, 0, 9, 9, -18, 18, -7, -9, -9, -9, 9, -4, -9, -9, 18, 9, 0, 18, 9, 0, -9, -9, -8, -9, 18, -9, 9, -18, 1, -9, -18, 9, 0, 18, 18, 0, 0, 9, -9, 18, -9, 5, -9, 0, -9, -9, -9, -18, 11, 9

Offset: 0

Paul D. Hanna, Feb 16 2011

sign

			G.f. = 1 - 4*x + 2*x^2 + 9*x^3 - 9*x^4 - 2*x^5 - 5*x^7 + 9*x^8 + ... - _Michael Somos_, Jul 12 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A182819, A144613, A000203.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), this sequence (k=3), A282937 (k=5), A282942 (k=7).

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jul 12 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n,-sigma(3*m)*x^m/m)+x*O(x^n)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(eta(X)^4/eta(X^3),n)}

G.f.: E(x)^4/E(x^3) where E(x) = Product_{n>=1} (1-x^n). [From a formula by Joerg Arndt in A182819]
a(n) = -(1/n)*Sum_{k=1..n} sigma(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017
Expansion of E(x) * E(x*w) * E(x/w) in powers of x^3 where w = exp(2 Pi i / 3). - Michael Somos, Jul 12 2018

A088838 Numerator of the quotient sigma(3n)/sigma(n).

Original entry on oeis.org

4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 121, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 121, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 364, 4, 4, 13, 4, 4, 13, 4, 4, 40

Offset: 1

Labos Elemer, Nov 04 2003

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A007949, A038712, A088837, A088839, A088840, A080278 (denominator), A144613.

Cf. A001620, A214369.

A088838 := proc(n)
    numtheory[sigma](3*n)/numtheory[sigma](n) ;
    numer(%) ;
end proc:
seq(A088838(n),n=1..100) ; # R. J. Mathar, Nov 19 2017
seq((3^(2+padic:-ordp(n,3))-1)/2, n=1..100); # Robert Israel, Nov 19 2017

k=3; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]

a(n) = numerator(sigma(3*n)/sigma(n)) \\ Felix Fröhlich, Nov 19 2017

From Robert Israel, Nov 19 2017: (Start)
a(n) = (3^(2+A007949(n))-1)/2.
G.f.: Sum_{k>=0} (3^(k+2)-1)*(x^(3^k)+x^(2*3^k))/(2*(1-x^(3^(k+1)))). (End)
a(n) = sigma(3*n)/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A144613(n)/A000203(n)).
Sum_{k=1..n} a(k) ~ (3/log(3))*n*log(n) + (1/2 + 3*(gamma-1)/log(3))*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A080278(k) = 4*A214369 + 1 = 3.728614... . (End)

A319526 Square array read by antidiagonals upwards: T(n,k) = sigma(n*k), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 3, 4, 7, 4, 7, 12, 12, 7, 6, 15, 13, 15, 6, 12, 18, 28, 28, 18, 12, 8, 28, 24, 31, 24, 28, 8, 15, 24, 39, 42, 42, 39, 24, 15, 13, 31, 32, 60, 31, 60, 32, 31, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 42, 40, 63, 48, 91, 48, 63, 40, 42, 12, 28, 36, 72, 91, 90, 96, 96, 90, 91, 72, 36, 28

Offset: 1

Omar E. Pol, Sep 25 2018

			The corner of the square array begins:
A000203:    1,   3,   4,   7,   6,  12,   8,  15,  13,  18,  12,  28, ...
A062731:    3,   7,  12,  15,  18,  28,  24,  31,  39,  42,  36,  60, ...
A144613:    4,  12,  13,  28,  24,  39,  32,  60,  40,  72,  48,  91, ...
A193553:    7,  15,  28,  31,  42,  60,  56,  63,  91,  90,  84, 124, ...
A283118:    6,  18,  24,  42,  31,  72,  48,  90,  78,  93,  72, 168, ...
A224613:   12,  28,  39,  60,  72,  91,  96, 124, 120, 168, 144, 195, ...
A283078:    8,  24,  32,  56,  48,  96,  57, 120, 104, 144,  96, 224, ...
A283122:   15,  31,  60,  63,  90, 124, 120, 127, 195, 186, 180, 252, ...
A283123:   13,  39,  40,  91,  78, 120, 104, 195, 121, 234, 156, 280, ...
...

Index entries for sequences related to sigma(n)

First 9 rows (also first 9 columns) are A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.
Main diagonal gives A065764.
Cf. A000203, A003991, A216626, A319073.

Table[DivisorSigma[1, # k] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

T(n,k) = A000203(n*k).

T(n,k) = A000203(A003991(n,k)).

A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

Original entry on oeis.org

1, 14, 59, 190, 401, 914, 1499, 2632, 4113, 6424, 8645, 13284, 17023, 23092, 30715, 40484, 48711, 63890, 75351, 95792, 116421, 139822, 159911, 199176, 229499, 267438, 309283, 364462, 404933, 482792, 532553, 611208, 688593, 772540, 862471, 998760, 1083615, 1200328

Offset: 1

Vaclav Kotesovec, May 10 2024

nonn

Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^4 for n = 1..100000

Cf. A069097, A372633, A372674.
Cf. A024916, A326124.
Cf. A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.

Table[Sum[DivisorSigma[1, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[1, n^2] + 2*Sum[DivisorSigma[1, j*n], {j, 1, n - 1}], {n, 2, 50}]]

a(n) ~ c * n^4, where c = Pi^4 / (144*zeta(3)) = 0.56274...

A341623 Numbers k such that sigma(3k) = 8k.

Original entry on oeis.org

28, 90, 496, 546, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176

Offset: 1

Antti Karttunen, Feb 19 2021

Every perfect number P greater than 6 (so, P is not divisible by 3) will be found in this sequence. Proof: sigma(3*P) = sigma(3)*sigma(P) = 4*(2*P) = 8*P. - Timothy L. Tiffin, Aug 26 2021

Solutions are integers y/3 where sigma(y)/y = 8/3. - Michel Marcus, Aug 27 2021

			546 is a term, since sigma(3*546) = sigma(1638) = 4368 = 8*546. - _Timothy L. Tiffin_, Aug 26 2021

Cf. A000396 (subsequence, apart from its terms that are divisible by 3).
Subsequence of A005101 and A227303.
Cf. also A000203, A144613, A326051, A336702, A347261.

Select[Range[5*10^9], DivisorSigma[1, 3*#] == 8*# &] (* Timothy L. Tiffin, Aug 26 2021 *)
Do[If[DivisorSigma[1, 3*k] == 8*k, Print[k]], {k, 5*10^9}] (* Timothy L. Tiffin, Aug 26 2021 *)

a(7)-a(8) from Martin Ehrenstein, Mar 06 2021

a(9)-a(10) from Michel Marcus, Aug 27 2021

cp's OEIS Frontend

A283123 a(n) = sigma(9*n).

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A283118 a(n) = sigma(5*n).

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A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

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A088838 Numerator of the quotient sigma(3n)/sigma(n).

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A319526 Square array read by antidiagonals upwards: T(n,k) = sigma(n*k), n >= 1, k >= 1.

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A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

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A341623 Numbers k such that sigma(3*k) = 8*k.

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A341623 Numbers k such that sigma(3k) = 8k.