A060640 -id:A060640 - cp's OEIS frontend

A127108 Triangle read by rows, A127099 * A000012.

1, 5, 2, 7, 3, 3, 17, 10, 4, 4, 11, 5, 5, 5, 5, 35, 23, 15, 6, 6, 6, 15, 7, 7, 7, 7, 7, 7, 49, 34, 20, 20, 8, 8, 8, 8, 34, 21, 21, 9, 9, 9, 9, 9, 9, 55, 37, 25, 25, 25, 10, 10, 10, 10, 10, 23, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 119, 91, 67, 46, 30, 30, 12, 12, 12, 12, 12, 12

Offset: 0

Gary W. Adamson, Jan 05 2007, Jul 27 2008

The operation A000012 * A127099 generates n-th row of the triangle by taking partial sums of n-th row of triangle A127099. Row 4 of A127099 (7, 6, 0, 4) becomes row 4 of A127108: (17, 10, 4, 4).
Row sums = A001001: (1, 7, 13, 35, 31, 91, ...).
Left column of the triangle = A060640: (1, 5, 7, 17, 11, 35, ...).

			First few rows of the triangle:
   1;
   5,  2;
   7,  3,  3;
  17, 10,  4,  4;
  11,  5,  5,  5,  5;
  35, 23, 15,  6,  6,  6;
  15,  7,  7,  7,  7,  7,  7;
  49, 34, 20, 20,  8,  8,  8,  8;
  34, 21, 21,  9,  9,  9,  9,  9,  9;
  55, 37, 25, 25, 25, 10, 10, 10, 10, 10;
  ...

Cf. A060640, A001001, A126988, A127093, A000203, A127094, A123229, A127096, A127098, A127099.

Triangle read by rows, A127099 * A000012.

Edited by N. J. A. Sloane, Aug 13 2008 at the suggestion of R. J. Mathar

A127168 Triangle read by rows: square of A126988.

Original entry on oeis.org

1, 4, 1, 6, 0, 1, 12, 4, 0, 1, 10, 0, 0, 0, 1, 24, 6, 4, 0, 0, 1, 14, 0, 0, 0, 0, 0, 1, 32, 12, 0, 4, 0, 0, 0, 1, 27, 0, 6, 0, 0, 0, 0, 0, 1, 40, 10, 0, 0, 4, 0, 0, 0, 0, 1, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 72, 24, 12, 6, 0, 4, 0, 0, 0, 0, 0, 1, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Gary W. Adamson, Jan 06 2007

Row sums = A060640: (1, 5, 7, 17, 11, 35, 15, 49...) Left column = A038040, d(n)*n: (1, 4, 6, 12, 10, 24, 14, 32, 27, 40...). A127168 * A008683 = A018804, (1, 3, 5, 8, 9, 15...); where A008683 = the Mobius sequence.

			First few rows of the triangle are:
1;
4, 1;
6, 0, 1;
12, 4, 0, 1
10, 0, 0, 0, 1
24, 6, 4, 0, 0, 1
14, 0, 0, 0, 0, 0, 1;
32, 12, 0, 4, 0, 0, 0, 1;
...

Cf. A008683, A018804, A038040, A060640, A127169.

a(20) = 1 inserted and more terms from Georg Fischer, May 31 2023

A143313 Triangle read by rows, A130540 * A000012, 1<=k<=n.

Original entry on oeis.org

1, 4, 1, 5, 1, 1, 11, 4, 1, 1, 7, 1, 1, 1, 1, 20, 8, 4, 1, 1, 19, 1, 1, 1, 1, 1, 1, 26, 11, 4, 4, 1, 1, 1, 1, 18, 5, 5, 1, 1, 1, 1, 1, 1, 28, 10, 4, 4, 4, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 55, 27, 15, 8, 4, 4, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Aug 06 2008

Left border = A007429: (1, 4, 5, 11, 7, 20, 9,...).

Row sums = A060640: (1, 5, 7, 17, 11, 35,...).

			First few rows of the triangle =
1;
4, 1;
5, 1, 1;
11, 4, 1, 1;
7, 1, 1, 1, 1;
20, 8, 4, 1, 1, 1;
9, 1, 1, 1, 1, 1, 1;
...
Row 4 = (11, 4, 1, 1) since row 4 of A130540 = (7, 3, 0, 1).

Cf. A130540, A007429, A060640.

Triangle read by rows, A130540 * A000012, 1<=k<=n. Equals partial row sums of A130540 starting from the right.

A306705 a(n) = Product_{d|n} d*tau(d), where tau(k) = the number of the divisors of k (A000005).

Original entry on oeis.org

1, 4, 6, 48, 10, 576, 14, 1536, 162, 1600, 22, 497664, 26, 3136, 3600, 122880, 34, 1679616, 38, 2304000, 7056, 7744, 46, 3057647616, 750, 10816, 17496, 6322176, 58, 3317760000, 62, 23592960, 17424, 18496, 19600, 470184984576, 74, 23104, 24336, 23592960000, 82

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

			a(6) = 1*tau(1) * 2*tau(2) * 3*tau(3) * 6*tau(6) = (1*1) * (2*2) * (3*2) * (6*4) = 576.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A060640 (Sum_{d|n} d*tau(d)), A007955, A211776.

[&*[d * NumberOfDivisors(d): d in Divisors(n)]: n in [1..100]]

f:= proc(n) uses numtheory; local d;
  mul(d*tau(d),d = divisors(n))
end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2019

Table[n^(DivisorSigma[0, n]/2) * Product[DivisorSigma[0, k], {k, Divisors[n]}], {n, 1, 60}] (* Vaclav Kotesovec, Mar 10 2019 *)

a(n) = my(res = 1); fordiv(n, d, res *= d*numdiv(d)); res; \\ Michel Marcus, Mar 06 2019

a(p) = 2p for p = primes (A000040).
a(n) = (Product_{d|n} tau(d)) * (Product_{d|n} d) = A211776(n) * A007955(n).
From Robert Israel, Mar 24 2019: (Start)
a(p^k) = (k+1)! * p^(k*(k+1)/2) for primes p.
a(p*q) = 16*p^2*q^2 if p and q are distinct primes. (End)

A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

Original entry on oeis.org

1, 1, 5, 1, 7, 7, 1, 11, 13, 17, 1, 19, 31, 35, 11, 1, 35, 85, 95, 31, 35, 1, 67, 247, 311, 131, 91, 15, 1, 131, 733, 1127, 631, 341, 57, 49, 1, 259, 2191, 4295, 3131, 1615, 351, 155, 34, 1, 515, 6565, 16775, 15631, 8645, 2409, 775, 130, 55, 1, 1027, 19687, 66311, 78131, 49111, 16815, 4991, 850, 217, 23

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
   1,   1,    1,     1,     1,      1,  ...
   5,   7,   11,    19,    35,     67,  ...
   7,  13,   31,    85,   247,    733,  ...
  17,  35,   95,   311,  1127,   4295,  ...
  11,  31,  131,   631,  3131,  15631,  ...
  35,  91,  341,  1615,  8645,  49111,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Columns k=0..3 give A060640, A001001, A027847, A027848.

Cf. A109974, A320940 (diagonal), A321876, A322103.

Table[Function[k, Sum[d DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, d^(k+1)*sigma(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} j*sigma_k(j)*x^j/(1 - x^j).
L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^sigma_k(j)).
A(n,k) = Sum_{d|n} d^(k+1)*sigma_1(n/d).

A340850 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-2) / (zeta(s-1))^2.

Original entry on oeis.org

1, 1, 4, 5, 16, 4, 36, 21, 40, 16, 100, 20, 144, 36, 64, 85, 256, 40, 324, 80, 144, 100, 484, 84, 416, 144, 364, 180, 784, 64, 900, 341, 400, 256, 576, 200, 1296, 324, 576, 336, 1600, 144, 1764, 500, 640, 484, 2116, 340, 1800, 416, 1024, 720, 2704, 364, 1600, 756, 1296, 784

Offset: 1

Werner Schulte, Jan 24 2021

Cf. A000290, A001157, A002618, A007433, A018804, A023900, A060640, A328722.

f[p_, e_] := (p^(2*e) - 1)*(p - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 24 2021 *)

Multiplicative with a(1) = 1 and a(p^e) = (p^(2*e)-1) * (p-1) / (p+1) for prime p and e > 0.
Dirichlet convolution of A002618 and A023900.
Dirichlet convolution of A001157 and A328722.
Dirichlet inverse b(n) for n > 0 is multiplicative with b(1) = 1 and b(p^e) = -(p-1)^2 * e * p^(e-1) for prime p and e > 0.
Dirichlet convolution with A060640 equals A007433.
Dirichlet convolution with A018804 equals A000290.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 12*zeta(3)/Pi^4 = 0.148083... . - Amiram Eldar, Oct 16 2022

A344787 a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 7, 31, 287, 3131, 47527, 823551, 16843583, 387440266, 10009772937, 285311670623, 8918294639219, 302875106592267, 11112685050294387, 437893920912795941, 18447025553014982271, 827240261886336764195, 39346558271492953948522, 1978419655660313589123999

Offset: 1

Wesley Ivan Hurt, May 28 2021

nonn

If p is prime, a(p) = p * Sum_{d|p} sigma_d(d) / d = p * (1 + (1^p + p^p)/p) = 1 + p + p^p.

			a(4) = 4 * Sum_{d|4} sigma_d(d) / d = 4 * ((1^1)/1 + (1^2 + 2^2)/2 + (1^4 + 2^4 + 4^4)/4) = 287.

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A245466, A321141, A334874, A343781, A344434, A344480.

Cf. A001001, A007429, A027847, A027848, A060640.

Table[n*Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k])/k, {k, n}], {n, 20}]

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Dec 16 2022

G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k)^2. - Seiichi Manyama, Dec 16 2022

A360429 Inverse Mobius transformation of A034714.

Original entry on oeis.org

1, 9, 19, 57, 51, 171, 99, 313, 262, 459, 243, 1083, 339, 891, 969, 1593, 579, 2358, 723, 2907, 1881, 2187, 1059, 5947, 1926, 3051, 3178, 5643, 1683, 8721, 1923, 7737, 4617, 5211, 5049, 14934, 2739, 6507, 6441, 15963, 3363, 16929, 3699, 13851, 13362, 9531, 4419, 30267, 7302, 17334

Offset: 1

R. J. Mathar, Feb 07 2023

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

Cf. A000005, A000012, A001620, A034714, A060640.

A360429 := proc(n)
    add(numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
end proc:

f[p_, e_] := ((e+1)*p^(2*e+4) - (e+2)*p^(2*e+2) + 1)/(p^2-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

a(n) = Sum_{d|n} A000005(d)*d^2.
Dirichlet convolution of A034714 and A000012.
Dirichlet g.f.: zeta^2(s-2)*zeta(s).
From Amiram Eldar, Feb 09 2023: (Start)
Multiplicative with a(p^e) = ((e+1)*p^(2*e+4) - (e+2)*p^(2*e+2) + 1)/(p^2-1)^2.
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 + zeta'(3)/zeta(3)) * n^3 * zeta(3)/3, where gamma is Euler's constant (A001620). (End)

A127572 Triangle, T(n,k) = sigma(k) * n/k if k|n, T(n,k) = 0 otherwise.

Original entry on oeis.org

1, 2, 3, 3, 0, 4, 4, 6, 0, 7, 5, 0, 0, 0, 6, 6, 9, 8, 0, 0, 12, 7, 0, 0, 0, 0, 0, 8, 8, 12, 0, 14, 0, 0, 0, 15, 9, 0, 12, 0, 0, 0, 0, 0, 13, 10, 15, 0, 0, 12, 0, 0, 0, 0, 18

Offset: 1

Gary W. Adamson, Jan 19 2007

			First few rows of the triangle are:
1;
2, 3;
3, 0, 4;
4, 6, 0, 7;
5, 0, 0, 0, 6;
6, 9, 8, 0, 0, 12;
...

Main diagonal = sigma(n), A000203; row sums = A060640.

Cf. A126988.

A126988 * M, where M = an infinite matrix with sigma(n) in the main diagonal and the rest zeros.

Edited by Franklin T. Adams-Watters, Aug 24 2011

A344042 a(n) = n * Sum_{d|n} sigma(d)^2 / d.

Original entry on oeis.org

1, 11, 19, 71, 41, 209, 71, 367, 226, 451, 155, 1349, 209, 781, 779, 1695, 341, 2486, 419, 2911, 1349, 1705, 599, 6973, 1166, 2299, 2278, 5041, 929, 8569, 1055, 7359, 2945, 3751, 2911, 16046, 1481, 4609, 3971, 15047, 1805, 14839, 1979, 11005, 9266, 6589, 2351, 32205, 3746, 12826, 6479

Offset: 1

Seiichi Manyama, May 08 2021

Cf. A060640, A065018, A226564, A344043.

a[n_] := n * DivisorSum[n, DivisorSigma[1, #]^2/# &]; Array[a, 51] (* Amiram Eldar, May 08 2021 *)

PARI
```
a(n) = n*sumdiv(n, d, sigma(d)^2/d);
```

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^2*x^k/(1-x^k)^2))

for(n=1, 100, print1(direuler(p=2, n, (1 - p^2*X^2) / ((1 - X) * (1 - p*X)^3 * (1 - p^2*X)))[n], ", ")) \\ Vaclav Kotesovec, May 08 2021

G.f.: Sum_{k >= 1} sigma(k)^2 * x^k/(1 - x^k)^2.
From Vaclav Kotesovec, May 08 2021: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1)^3 * zeta(s-2) / zeta(2*s-2).
Sum_{k=1..n} a(k) ~ 5 * Pi^2 * zeta(3) * n^3 / 36. (End)

cp's OEIS Frontend

A127108 Triangle read by rows, A127099 * A000012.

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A127168 Triangle read by rows: square of A126988.

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A143313 Triangle read by rows, A130540 * A000012, 1<=k<=n.

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A306705 a(n) = Product_{d|n} d*tau(d), where tau(k) = the number of the divisors of k (A000005).

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A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

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A340850 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-2) / (zeta(s-1))^2.

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A344787 a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.

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A360429 Inverse Mobius transformation of A034714.

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