A035116 -id:A035116 - cp's OEIS frontend

A356533 a(n) = sigma_2(n)^2.

1, 25, 100, 441, 676, 2500, 2500, 7225, 8281, 16900, 14884, 44100, 28900, 62500, 67600, 116281, 84100, 207025, 131044, 298116, 250000, 372100, 280900, 722500, 423801, 722500, 672400, 1102500, 708964, 1690000, 925444, 1863225, 1488400, 2102500, 1690000, 3651921

Offset: 1

Vaclav Kotesovec, Aug 11 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (15), a=b=2.

Cf. A001157, A127473, A035116, A072861, A356535 (partial sums).

Mathematica
```
Table[DivisorSigma[2, n]^2, {n, 1, 40}]
```

a(n) = sigma(n, 2)^2; \\ Michel Marcus, Aug 11 2022

Dirichlet g.f.: zeta(s) * zeta(s-2)^2 * zeta(s-4) / zeta(2*s-4).
Multiplicative with a(p^e) = ((p^(2*e+2)-1)/(p^2-1))^2. - Amiram Eldar, Aug 11 2022
a(n) = A001157(n)^2. - R. J. Mathar, Aug 18 2022

A356534 a(n) = sigma_3(n)^2.

Original entry on oeis.org

1, 81, 784, 5329, 15876, 63504, 118336, 342225, 573049, 1285956, 1774224, 4177936, 4831204, 9585216, 12446784, 21911761, 24147396, 46416969, 47059600, 84603204, 92775424, 143712144, 148060224, 268304400, 248094001, 391327524, 417793600, 630612544, 594872100

Offset: 1

Vaclav Kotesovec, Aug 11 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (15), a=b=3.

Cf. A001158, A127473, A035116, A072861, A356536 (partial sums).

Mathematica
```
Table[DivisorSigma[3, n]^2, {n, 1, 40}]
```

a(n) = sigma(n, 3)^2; \\ Michel Marcus, Aug 11 2022

Dirichlet g.f.: zeta(s) * zeta(s-3)^2 * zeta(s-6) / zeta(2*s-6).

Multiplicative with a(p^e) = ((p^(3*e+3)-1)/(p^3-1))^2. - Amiram Eldar, Aug 11 2022

A356574 a(n) = Sum_{d|n} tau(d^4), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 6, 6, 15, 6, 36, 6, 28, 15, 36, 6, 90, 6, 36, 36, 45, 6, 90, 6, 90, 36, 36, 6, 168, 15, 36, 28, 90, 6, 216, 6, 66, 36, 36, 36, 225, 6, 36, 36, 168, 6, 216, 6, 90, 90, 36, 6, 270, 15, 90, 36, 90, 6, 168, 36, 168, 36, 36, 6, 540, 6, 36, 90, 91, 36, 216, 6, 90, 36, 216, 6, 420, 6, 36, 90, 90

Offset: 1

Seiichi Manyama, Dec 13 2022

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

Cf. A007425, A035116, A061391, A358380, A359037, A359038.

Cf. A000005, A321348.

Array[DivisorSum[#, DivisorSigma[0, #^4] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 2*e^2 + 3*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)

PARI
```
a(n) = sumdiv(n, d, numdiv(d^4));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n*d^2));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n^2));
```
PARI
```
a(n) = numdiv(n)*numdiv(n^2);
```

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^4)*x^k/(1-x^k)))

from math import prod
from sympy import factorint
def A356574(n): return prod((e+1)*((e<<1)+1) for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

a(n) = Sum_{d|n} tau(n * d^2) = Sum_{d|n} tau(n^2).
a(n) = tau(n) * tau(n^2).
G.f.: Sum_{k>=1} tau(k^4) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 2*e^2 + 3*e + 1. - Amiram Eldar, Dec 14 2022

A359037 a(n) = Sum_{d|n} tau(d^6), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 8, 8, 21, 8, 64, 8, 40, 21, 64, 8, 168, 8, 64, 64, 65, 8, 168, 8, 168, 64, 64, 8, 320, 21, 64, 40, 168, 8, 512, 8, 96, 64, 64, 64, 441, 8, 64, 64, 320, 8, 512, 8, 168, 168, 64, 8, 520, 21, 168, 64, 168, 8, 320, 64, 320, 64, 64, 8, 1344, 8, 64, 168, 133, 64, 512, 8, 168, 64, 512, 8, 840, 8

Offset: 1

Seiichi Manyama, Dec 13 2022

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

Cf. A007425, A035116, A061391, A356574, A358380, A359038.

Cf. A000005, A321348.

Array[DivisorSum[#, DivisorSigma[0, #^6] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 3*e^2 + 4*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)

PARI
```
a(n) = sumdiv(n, d, numdiv(d^6));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n*d^4));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n^2*d^2));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n^3));
```
PARI
```
a(n) = numdiv(n)*numdiv(n^3);
```

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^6)*x^k/(1-x^k)))

a(n) = Sum_{d|n} tau(n * d^4) = Sum_{d|n} tau(n^2 * d^2) = Sum_{d|n} tau(n^3).
a(n) = tau(n) * tau(n^3).
G.f.: Sum_{k>=1} tau(k^6) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 3*e^2 + 4*e + 1. - Amiram Eldar, Dec 14 2022

A359038 a(n) = Sum_{d|n} tau(d^7), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 9, 9, 24, 9, 81, 9, 46, 24, 81, 9, 216, 9, 81, 81, 75, 9, 216, 9, 216, 81, 81, 9, 414, 24, 81, 46, 216, 9, 729, 9, 111, 81, 81, 81, 576, 9, 81, 81, 414, 9, 729, 9, 216, 216, 81, 9, 675, 24, 216, 81, 216, 9, 414, 81, 414, 81, 81, 9, 1944, 9, 81, 216, 154, 81, 729, 9, 216, 81, 729, 9

Offset: 1

Seiichi Manyama, Dec 13 2022

Cf. A000005, A007425, A035116, A061391, A356574, A358380, A359037.

Cf. A321348.

Array[DivisorSum[#, DivisorSigma[0, #^7] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 7*e^2/2 + 9*e/2 + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)

PARI
```
a(n) = sumdiv(n, d, numdiv(d^7));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n*d^5));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n^2*d^3));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n^3*d));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n^4/d));
```

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^7)*x^k/(1-x^k)))

from math import prod
from sympy import factorint
def A359038(n): return prod((e+1)*(7*e+2)>>1 for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

a(n) = Sum_{d|n} tau(n * d^5) = Sum_{d|n} tau(n^2 * d^3) = Sum_{d|n} tau(n^3 * d) = Sum_{d|n} tau(n^4 / d).
G.f.: Sum_{k>=1} tau(k^7) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 7*e^2/2 + 9*e/2 + 1. - Amiram Eldar, Dec 14 2022

A143235 Triangle read by rows: T(n,k) = tau(n)*tau(k), the product of the number of divisors.

Original entry on oeis.org

1, 2, 4, 2, 4, 4, 3, 6, 6, 9, 2, 4, 4, 6, 4, 4, 8, 8, 12, 8, 16, 2, 4, 4, 6, 4, 8, 4, 4, 8, 8, 12, 8, 16, 8, 16, 3, 6, 6, 9, 6, 12, 6, 12, 9, 4, 8, 8, 12, 8, 16, 8, 16, 12, 16, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 6, 12, 12, 18, 12, 24, 12, 24, 18, 24, 12, 36, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4

Offset: 1

Gary W. Adamson, Aug 01 2008

The triangle can also be created by the triple matrix product A130209 * A000012 * A130209.

			First few rows of the triangle =
  1;
  2, 4;
  2, 4, 4;
  3, 6, 6,  9;
  2, 4, 4,  6, 4;
  4, 8, 8, 12, 8, 16;
  2, 4, 4,  6, 4,  8, 4;
  4, 8, 8, 12, 8, 16, 8, 16;
  3, 6, 6,  9, 6, 12, 6, 12, 9;
  ...
T(9,6) = 12 = d(9)*d(6) = 3*4.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000005, A035116 (right diagonal), A143236 (row sums).

A143235:= func< n,k | NumberOfDivisors(n)*NumberOfDivisors(k) >;
[A143235(n,k): k in [1..n], n in [1..14]]; // G. C. Greubel, Sep 12 2024

A143235[n_, k_]:= DivisorSigma[0, n]*DivisorSigma[0, k];
Table[A143235[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Sep 12 2024 *)

def A143235(n,k): return sigma(n,0)*sigma(k,0)
flatten([[A143235(n,k) for k in range(1,n+1)] for n in range(1,15)]) # G. C. Greubel, Sep 12 2024

T(n,k) = A000005(n)*A000005(k), for 1 <= k <= n, n >= 1.

Sum_{k=1..n} T(n, k) = A143236(n) (row sums).

A337175 Number of pairs of divisors of n, (d1,d2), such that d1 <= d2 and d1*d2 < n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 4, 2, 4, 1, 9, 1, 4, 4, 6, 1, 9, 1, 9, 4, 4, 1, 16, 2, 4, 4, 9, 1, 16, 1, 9, 4, 4, 4, 20, 1, 4, 4, 16, 1, 16, 1, 9, 9, 4, 1, 25, 2, 9, 4, 9, 1, 16, 4, 16, 4, 4, 1, 36, 1, 4, 9, 12, 4, 16, 1, 9, 4, 16, 1, 36, 1, 4, 9, 9, 4, 16, 1, 25, 6, 4, 1, 36, 4, 4

Offset: 1

Wesley Ivan Hurt, Jan 28 2021

			a(9) = 2; (1,1), (1,3).
a(10) = 4; (1,1), (1,2), (1,5), (2,2).
a(11) = 1; (1,1).
a(12) = 9; (1,1), (1,2), (1,3), (1,4), (1,6), (2,2), (2,3), (2,4), (3,3).

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

Cf. A000005, A002620, A038548, A337174, A035116, A010052.

Table[Sum[Sum[(1 - Sign[Floor[i*k/n]]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 80}]
a[n_] := Floor[DivisorSigma[0, n]^2/4]; Array[a, 100] (* Amiram Eldar, Feb 01 2025 *)

a(n) = numdiv(n)^2\4; \\ Amiram Eldar, Feb 01 2025

from sympy import divisor_count
def A337175(n):
    return divisor_count(n)**2//4  # Chai Wah Wu, Jan 29 2021

a(n) = Sum_{d1|n, d2|n} (1 - sign(floor(d1*d2/n))).
a(n) = tau^2/4 if tau is even and a(n) = (tau-1)*(tau+1)/4 if tau is odd, where tau = A000005(n) is the number of divisors of n, i.e., a(n) = A002620(A000005(n)) = floor(A000005(n)^2/4). - Chai Wah Wu, Jan 29 2021
a(n) = (A035116(n) - A010052(n))/4. - Ridouane Oudra, May 20 2025
a(n) = A337174(n) - A038548(n). - Ridouane Oudra, May 28 2025

A356535 a(n) = Sum_{k=1..n} sigma_2(k)^2.

Original entry on oeis.org

1, 26, 126, 567, 1243, 3743, 6243, 13468, 21749, 38649, 53533, 97633, 126533, 189033, 256633, 372914, 457014, 664039, 795083, 1093199, 1343199, 1715299, 1996199, 2718699, 3142500, 3865000, 4537400, 5639900, 6348864, 8038864, 8964308, 10827533, 12315933, 14418433

Offset: 1

Vaclav Kotesovec, Aug 11 2022

nonn

Partial sums of A356533.

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

Cf. A001157, A057434, A061502, A072379, A356536.

Cf. A127473, A035116, A072861, A356533, A356534.

Table[Sum[DivisorSigma[2, k]^2, {k, 1, n}], {n, 1, 40}]

a(n) = sum(k=1, n, sigma(k, 2)^2); \\ Michel Marcus, Aug 11 2022

a(n) ~ 189 * zeta(3)^2 * zeta(5) * n^5 / Pi^6.

A356536 a(n) = Sum_{k=1..n} sigma_3(k)^2.

Original entry on oeis.org

1, 82, 866, 6195, 22071, 85575, 203911, 546136, 1119185, 2405141, 4179365, 8357301, 13188505, 22773721, 35220505, 57132266, 81279662, 127696631, 174756231, 259359435, 352134859, 495847003, 643907227, 912211627, 1160305628, 1551633152, 1969426752, 2600039296

Offset: 1

Vaclav Kotesovec, Aug 11 2022

nonn

Partial sums of A356534.

In general, for m>0, Sum_{k=1..n} sigma_m(k)^2 ~ zeta(2*m+1) * zeta(m+1)^2 * n^(2*m+1) / ((2*m+1) * zeta(2*m+2)).

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

Cf. A001158, A057434, A061502, A072379, A356535.

Cf. A127473, A035116, A072861, A356533, A356534.

Table[Sum[DivisorSigma[3, k]^2, {k, 1, n}], {n, 1, 40}]
Accumulate[DivisorSigma[3,Range[40]]^2] (* This program is much more efficient than the first program above. *) (* Harvey P. Dale, Feb 27 2023 *)

a(n) = sum(k=1, n, sigma(k, 3)^2); \\ Michel Marcus, Aug 11 2022

a(n) ~ zeta(7) * n^7 / 6.

A358380 a(n) = Sum_{d|n} tau(d^5), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 7, 7, 18, 7, 49, 7, 34, 18, 49, 7, 126, 7, 49, 49, 55, 7, 126, 7, 126, 49, 49, 7, 238, 18, 49, 34, 126, 7, 343, 7, 81, 49, 49, 49, 324, 7, 49, 49, 238, 7, 343, 7, 126, 126, 49, 7, 385, 18, 126, 49, 126, 7, 238, 49, 238, 49, 49, 7, 882, 7, 49, 126, 112, 49, 343, 7, 126, 49, 343, 7, 612, 7, 49, 126, 126

Offset: 1

Seiichi Manyama, Dec 13 2022

Cf. A000005, A007425, A035116, A061391, A356574, A359037, A359038.

Cf. A321348.

Array[DivisorSum[#, DivisorSigma[0, #^5] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 5*e^2/2 + 7*e/2 + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)

PARI
```
a(n) = sumdiv(n, d, numdiv(d^5));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n*d^3));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n^2*d));
```
PARI
```
a(n) = sumdiv(n, d, numdiv(n^3/d));
```

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^5)*x^k/(1-x^k)))

a(n) = Sum_{d|n} tau(n * d^3) = Sum_{d|n} tau(n^2 * d) = Sum_{d|n} tau(n^3 / d).
G.f.: Sum_{k>=1} tau(k^5) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 5*e^2/2 + 7*e/2 + 1. - Amiram Eldar, Dec 14 2022

cp's OEIS Frontend

A356533 a(n) = sigma_2(n)^2.

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A356534 a(n) = sigma_3(n)^2.

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A356574 a(n) = Sum_{d|n} tau(d^4), where tau(n) = number of divisors of n, cf. A000005.

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A359037 a(n) = Sum_{d|n} tau(d^6), where tau(n) = number of divisors of n, cf. A000005.

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A359038 a(n) = Sum_{d|n} tau(d^7), where tau(n) = number of divisors of n, cf. A000005.

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Mathematica

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A143235 Triangle read by rows: T(n,k) = tau(n)*tau(k), the product of the number of divisors.

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A337175 Number of pairs of divisors of n, (d1,d2), such that d1 <= d2 and d1*d2 < n.