A372966 -id:A372966 - cp's OEIS frontend

A373039 a(n) = (A372966(n) - 1)/240.

0, 1, 27, 257, 1625, 6508, 24010, 65793, 177174, 391626, 893101, 1665644, 3398759, 5786411, 10531652, 16843009, 29065308, 42698935, 70764303, 100231882, 155608837, 215237342, 326294606, 426404460, 634767250, 819100920, 1162438641, 1480961067, 2084357107, 2538128133

Offset: 1

Hugo Pfoertner, May 20 2024

nonn

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

Cf. A000290, A001159, A013956.

f[p_, e_] := (p^(8*e + 4) + 1)/(p^4 + 1); a[1] = 0; a[n_] := (Times @@ f @@@ FactorInteger[n] - 1) / 240; Array[a, 30] (* Amiram Eldar, Jan 08 2025 *)

a(n) = (sigma(n^2, 8)/sigma(n^2, 4)-1)/240

From Amiram Eldar, Jan 08 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-8)/zeta(s-4) - 1)/240.
Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/(2160*zeta(5)) = 0.000447372... . (End)

A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).

Original entry on oeis.org

1, 13, 73, 205, 601, 949, 2353, 3277, 5905, 7813, 14521, 14965, 28393, 30589, 43873, 52429, 83233, 76765, 129961, 123205, 171769, 188773, 279313, 239221, 375601, 369109, 478297, 482365, 706441, 570349, 922561, 838861, 1060033, 1082029

Offset: 1

Benoit Cloitre, Jun 21 2003

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001157, A001159, A002117, A007434, A013663, A065827.
Cf. A068963, A372962, A372963, A373007.
Cf. A057660, A084220, A372966, A373105.

with(numtheory): a:=n->sigma[4](n^2)/sigma[2](n^2): seq(a(n),n=1..40); # Muniru A Asiru, Oct 09 2018

Table[DivisorSigma[4, n^2]/DivisorSigma[2, n^2], {n, 1, 50}] (* G. C. Greubel, Oct 08 2018 *)
f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Sep 13 2020 *)

a(n)=sumdiv(n^2,d,d^4)/sumdiv(n^2,d,d^2)

a(n) = sigma(n^2, 4)/sigma(n^2, 2); \\ Michel Marcus, Oct 09 2018

Multiplicative with a(p^e) = (p^(4*e + 2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 13 2020
Sum_{k>=1} 1/a(k) = 1.09957644430375183822287768590764825667080036406680891521221069625517483696... - Vaclav Kotesovec, Sep 24 2020
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(3)) = 0.172525... . - Amiram Eldar, Oct 30 2022
From Peter Bala, Jan 18 2024: (Start)
a(n) = Sum_{d divides n} J_2(d^2) = Sum_{d divides n} d^2 * J_2(d), where the Jordan totient function J_2(n) = A007434(n).
a(n) = Sum_{1 <= j, k <= n} ( n/gcd(j, k, n) )^2.
Dirichlet g.f.: zeta(s) * zeta(s-4) / zeta(s-2) [Corrected by Michael Shamos, May 18 2025]. (End)
a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_4(d). - Seiichi Manyama, May 18 2024

A371491 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^3.

Original entry on oeis.org

1, 249, 6535, 63737, 390501, 1627215, 5764459, 16316665, 42876109, 97234749, 214357551, 416521295, 815728525, 1435350291, 2551924035, 4177066233, 6975752529, 10676151141, 16983556183, 24889362237, 37670739565, 53375030199, 78310973115, 106629405775

Offset: 1

Seiichi Manyama, May 24 2024

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

Cf. A068970, A084220, A372950, A372964.
Cf. A372952, A372963.
Cf. A372966.
Cf. A013664, A013667.

f[p_, e_] := (p^(8*e + 8) - p^(8*e + 3) + p^3 - 1)/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 24 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 8));

a(n) = sumdiv(n,d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 8)/sigma(d^2, 4));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_8(d).
a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_8(d^2)/sigma_4(d^2).
From Amiram Eldar, May 24 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e+8) - p^(8*e+3) + p^3 - 1)/(p^8-1).
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(9)/zeta(6) = 0.984926747... . (End)
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5. - Seiichi Manyama, May 25 2024

A373105 a(n) = sigma_10(n^2)/sigma_5(n^2).

Original entry on oeis.org

1, 993, 58807, 1016801, 9762501, 58395351, 282458443, 1041204193, 3472494301, 9694163493, 25937263551, 59795016407, 137858120557, 280481233899, 574103396307, 1066193093601, 2015992480593, 3448186840893, 6131063781703, 9926520779301

Offset: 1

Seiichi Manyama, May 25 2024

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

Cf. A057660, A084218, A084220, A372966.
Cf. A371491, A371878, A373007, A373103.
Cf. A013664, A013669.

f[p_, e_] := (p^(10*e+5) + 1)/(p^5 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, May 25 2024 *)

PARI
```
a(n) = sigma(n^2, 10)/sigma(n^2, 5);
```

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^5*sigma(d, 10));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x^5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^5 * sigma_10(d).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(10*e+5) + 1)/(p^5 + 1).
Dirichlet g.f.: zeta(s)*zeta(s-10)/zeta(s-5).
Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = zeta(11)/zeta(6) = 0.9834383562... . (End)

A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).

Original entry on oeis.org

1, 466, 9681, 123106, 487345, 4511346, 6722401, 31576546, 63779361, 227102770, 235777201, 1191789186, 883674961, 3132638866, 4717986945, 8084578786, 7411648321, 29721182226, 17926949521, 59995093570, 65079564081, 109872175666, 81870270241, 305692541826

Offset: 1

Seiichi Manyama, May 26 2024

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

Cf. A059377, A372966.
Cf. A062952, A373132, A373133.
Cf. A013663, A013667.

f[p_, e_] := (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=4, m=4) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{d|n} J_4(d) * sigma(d^4), where the Jordan totient function J_4(n) = A059377(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)).
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(5) * zeta(9) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 - 1/p^8 - 1/p^9 + 1/p^10) = 1.83382546873826519758... . (End)

A108223 a(n) = sigma_{2n}(n^2)/sigma_n(n^2), where sigma_n(m) = Sum_{d|m} d^n.

Original entry on oeis.org

1, 13, 703, 61681, 9762501, 2140365529, 678222249307, 280379743338241, 150087010086914941, 99902428887422922553, 81402749386554449442711, 79477293980103609858493681, 91733330193268313783293023757, 123469159731637675342948027295569, 191751045863140709562160603031808243

Offset: 1

Leroy Quet, Jun 28 2005

nonn

			sigma_4(4)/sigma_2(4) =
(1 + 2^4 + 4^4)/(1 + 2^2 + 4^2) = 13.

Cf. A062755.

Cf. A057660, A084218, A084220, A372966.

Table[DivisorSigma[2n, n^2]/DivisorSigma[n, n^2], {n, 10}] (* Ryan Propper, Apr 03 2007 *)

a(n) = sigma(n^2, 2*n)/sigma(n^2, n); \\ Michel Marcus, Sep 06 2019

a(n) = Product_{p=primes} (Sum_{k=0..2*b(n, p)} p^(n*k)*(-1)^k), where p^b(n, p) is the highest power of p dividing n.
From Seiichi Manyama, May 18 2024: (Start)
a(n) = Sum_{1 <= x_1, x_2, ... , x_n <= n} ( n/gcd(x_1, x_2, ... , x_n, n) )^n.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^n * sigma_{2*n}(d). (End)

More terms from Ryan Propper, Apr 03 2007

More terms from Michel Marcus, Sep 06 2019

A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.

Original entry on oeis.org

1, 17, 163, 529, 2501, 2771, 14407, 16913, 39529, 42517, 146411, 86227, 342733, 244919, 407663, 541201, 1336337, 671993, 2345779, 1323029, 2348341, 2488987, 6156503, 2756819, 7815001, 5826461, 9605467, 7621303, 19803869, 6930271, 27705631, 17318417, 23864993

Offset: 1

Seiichi Manyama, May 18 2024

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

Cf. A057660, A068963, A068970.
Cf. A001160, A008683.
Cf. A372966.
Cf. A013661, A013664, A157292.

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

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 5));

PARI
```
a(n) = sumdiv(n, d, eulerphi(d^5));
```

a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_5(d).
a(n) = Sum_{d|n} d^(5-m) * phi(d^m) for m > 0.
G.f.: Sum_{k>=1} k^(5-m) * phi(k^m) * x^k/(1 - x^k) for m > 0.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-4).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(2) = 2*Pi^4/315 = 0.6184704192... (1/A157292). (End)

A373040 a(n) = (A084218(n) - 1)/12.

Original entry on oeis.org

0, 1, 6, 17, 50, 79, 196, 273, 492, 651, 1210, 1247, 2366, 2549, 3656, 4369, 6936, 6397, 10830, 10267, 14314, 15731, 23276, 19935, 31300, 30759, 39858, 40197, 58870, 47529, 76880, 69905, 88336, 90169, 117846, 100877, 156066, 140791, 172724, 164123, 235340, 186083

Offset: 1

Hugo Pfoertner, May 20 2024

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

Cf. A000290, A001157, A001159, A084218, A372966, A373039.

Cf. A002117, A013663.

f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 0; a[n_] := (Times @@ f @@@ FactorInteger[n] - 1) / 12; Array[a, 35] (* Amiram Eldar, Jan 03 2025 *)

a(n) = (sigma(n^2, 4)/sigma(n^2, 2) - 1)/12

From Amiram Eldar, Jan 03 2025: (Start)
Dirichlet g.f.: (zeta(s-4)/zeta(s-2) - zeta(s))/12.
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(60*zeta(3)) = 0.0143771... . (End)

cp's OEIS Frontend

A373039 a(n) = (A372966(n) - 1)/240.

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A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).

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A371491 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^3.

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A373105 a(n) = sigma_10(n^2)/sigma_5(n^2).

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A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).

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A108223 a(n) = sigma_{2n}(n^2)/sigma_n(n^2), where sigma_n(m) = Sum_{d|m} d^n.

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A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.

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A373040 a(n) = (A084218(n) - 1)/12.

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