A077456 -id:A077456 - cp's OEIS frontend

A077454 a(n) = sigma_3(n^3)/sigma(n^3).

1, 39, 511, 2359, 12621, 19929, 101179, 149943, 368089, 492219, 1611831, 1205449, 4457701, 3945981, 6449331, 9588151, 22722609, 14355471, 44576623, 29772939, 51702469, 62861409, 141611691, 76620873, 196890121, 173850339, 268218727, 238681261, 574336533, 251523909

Offset: 1

Benoit Cloitre, Nov 30 2002

			a(2) = sigma_3(2^3)/sigma(2^3) = 585/15 = 39.

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

Cf. A000203, A000578, A001158, A013665, A057660, A077455, A077456.

f[p_, e_] := (p^(6*e+2) + p^(3*e+1) + 1)/(p^2 + p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 30] (* Amiram Eldar, Sep 09 2020 *)

PARI
```
a(n)=sumdiv(n^3,d,d^3)/sigma(n^3)
```

a(n) = my(f=factor(n^3)); sigma(f, 3)/sigma(f); \\ Michel Marcus, Sep 09 2020

a(n) = A001158(n^3)/A000203(n^3).
Multiplicative with a(p^e) = (p^(6*e+2) + p^(3*e+1) + 1)/(p^2 + p + 1). - Amiram Eldar, Sep 09 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)*Pi^4/630) * Product_{p prime} (1 - 1/p^2 - 1/p^6 + 1/p^7 - 1/p^8 + 1/p^9) = 0.09343400455... . - Amiram Eldar, Oct 28 2022

A077455 a(n) = sigma_4(n^4)/sigma(n^4).

Original entry on oeis.org

1, 2255, 360205, 8965359, 195688121, 812262275, 11869610005, 36654862063, 190649623129, 441276712855, 2853329308061, 3229367138595, 21506735660905, 26765970561275, 70487839624805, 150121132912367, 548357292625505, 429914900155895, 2096841596815405, 1754414256800439

Offset: 1

Benoit Cloitre, Nov 30 2002

			a(2) = sigma_4(2^4)/sigma(2^4) = 69905/31 = 2255.

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

Cf. A000203, A000583, A001158, A057660, A077454, A077456.

Cf. A002117, A013663, A013667, A013671.

f[p_, e_] := (p^(12*e+3) + p^(8*e+2) + p^(4*e+1) + 1)/(p^3 + p^2 + p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 20] (* Amiram Eldar, Sep 09 2020 *)

PARI
```
a(n)=sumdiv(n^4,d,d^4)/sigma(n^4)
```

a(n) = my(f=factor(n^4)); sigma(f, 4)/sigma(f); \\ Michel Marcus, Sep 09 2020

a(n) = A001158(n^4)/A000203(n^4).
Multiplicative with a(p^e) = (p^(12*e+3) + p^(8*e+2) + p^(4*e+1) + 1)/(p^3 + p^2 + p + 1). - Amiram Eldar, Sep 09 2020
Sum_{k=1..n} a(k) ~ c * n^13, where c = (zeta(3)*zeta(5)*zeta(9)*zeta(13)/13) * Product_{p prime} (1-1/p^2-1/p^3+1/p^5-1/p^7+1/p^8-1/p^12+2/p^13-2/p^14+2/p^15-1/p^16+2/p^17-3/p^18+1/p^19+1/p^21-1/p^22-1/p^26-1/p^27) = 0.048281563902... . - Amiram Eldar, Nov 20 2022

A077457 a(n) = sigma_4(n^4)/sigma_2(n^4).

Original entry on oeis.org

1, 205, 5905, 52429, 375601, 1210525, 5649505, 13421773, 38742049, 76998205, 212601841, 309593245, 810932305, 1158148525, 2217923905, 3435973837, 6951703105, 7942120045, 16936647121, 19692384829, 33360327025, 43583377405, 78163228705, 79255569565, 146719125601

Offset: 1

Benoit Cloitre, Nov 30 2002

sigma_y(n^x) divides sigma_x(n^x) for all n if y divides x.

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

Cf. A001157, A001159, A013667, A057660, A077454, A077455, A077456.

f[p_, e_] := (p^(8*e+2) + 1)/(p^2 + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 25] (* Amiram Eldar, Sep 09 2020 *)

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

a(n) = my(f=factor(n^4)); sigma(f, 4)/sigma(f, 2); \\ Michel Marcus, Sep 09 2020

a(n) = A001159(n^4)/A001157(n^4).
Multiplicative with a(p^e) = (p^(8*e+2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 09 2020
Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)/9) * Product_{p prime} (1 - 1/p^3 + 1/p^5 - 1/p^7) = 0.09549806119... . - Amiram Eldar, Oct 28 2022

cp's OEIS Frontend

A077454 a(n) = sigma_3(n^3)/sigma(n^3).

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A077455 a(n) = sigma_4(n^4)/sigma(n^4).

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Formula

A077457 a(n) = sigma_4(n^4)/sigma_2(n^4).

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