A344044 -id:A344044 - cp's OEIS frontend

A344043 a(n) = n * Sum_{d|n} sigma(d)^3 / d.

1, 29, 67, 401, 221, 1943, 519, 4177, 2398, 6409, 1739, 26867, 2757, 15051, 14807, 38145, 5849, 69542, 8019, 88621, 34773, 50431, 13847, 279859, 30896, 79953, 71194, 208119, 27029, 429403, 32799, 326337, 116513, 169621, 114699, 961598, 54909, 232551, 184719, 923117, 74129, 1008417, 85227, 697339

Offset: 1

Seiichi Manyama, May 08 2021

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

Cf. A002117, A060640, A226565, A226566, A344044.

a[n_] := n * DivisorSum[n, DivisorSigma[1, #]^3/# &]; Array[a, 44] (* Amiram Eldar, May 08 2021 *)

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

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^3*x^k/(1-x^k)^2))

G.f.: Sum_{k >= 1} sigma(k)^3 * x^k/(1 - x^k)^2.

Sum_{k=1..n} a(k) ~ c * n^4, where c = (Pi^6*zeta(3)^2/2160) * Product_{p prime} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 1.8238925519... . - Amiram Eldar, Nov 20 2022

A344047 a(n) = Sum_{d|n} sigma(d)^d.

Original entry on oeis.org

1, 10, 65, 2411, 7777, 2986058, 2097153, 2562893036, 10604499438, 3570467234410, 743008370689, 232218265092200875, 793714773254145, 21035720123170684938, 504857282956046114465, 727423121747187826721517, 2185911559738696531969, 43567528752021332763809905512, 5242880000000000000000001

Offset: 1

Seiichi Manyama, May 08 2021

nonn

Cf. A065018, A342473, A344044.

a[n_] := DivisorSum[n, DivisorSigma[1, #]^# &]; Array[a, 19] (* Amiram Eldar, May 08 2021 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (sigma(k)*x)^k/(1-x^k)))

G.f.: Sum_{k >= 1} (sigma(k) * x)^k/(1 - x^k).

If p is prime, a(p) = 1 + (p+1)^p.

A226565 Numbers k such that Sum_{d|k} sigma(d)^3 is a multiple of k.

Original entry on oeis.org

1, 2, 14, 32, 39, 42, 78, 96, 105, 117, 126, 133, 189, 195, 210, 224, 234, 266, 288, 378, 390, 399, 465, 480, 546, 585, 672, 793, 798, 930, 975, 1170, 1197, 1248, 1365, 1470, 1586, 1638, 1862, 1950, 1995, 2016, 2379, 2394, 2646, 2730, 3255, 3360, 3393, 3591

Offset: 1

Paolo P. Lava, Jun 11 2013

nonn

			Divisors of 189 are 1, 3, 7, 9, 21, 27, 63, 189, sigma(1) = 1, sigma(3) = 4, sigma(7) = 8, sigma(9) = 13, sigma(21) = 32, sigma(27) = 40, sigma(63) = 104, sigma(189) = 320. (1^3 + 4^3 + 8^3 + 13^3 + 32^3 + 40^3 + 104^3 + 320^3) / 189 = 179854.

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..100 from Paolo P. Lava)

Cf. A068978, A068986, A226563, A226564, A226566, A344044.

with(numtheory); ListA226565:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^3/n,k=1..nops(a));
if type(b,integer) then print(n); fi; od; end: ListA226565 (10^6);

Select[Range[4000],Divisible[Total[DivisorSigma[1,#]^3&/@Divisors[#]],#]&] (* Harvey P. Dale, Sep 17 2019 *)
s[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Select[Range[3600], Divisible[s[#], #] &] (* Amiram Eldar, Jul 01 2022 *)

A344060 a(n) = Sum_{d|n} sigma(d)^n.

Original entry on oeis.org

1, 10, 65, 2483, 7777, 2990810, 2097153, 2568661988, 10604761518, 3570527751850, 743008370689, 232227195048256531, 793714773254145, 21035724521219881850, 504857283427304833025, 727429690188773950335429, 2185911559738696531969, 43567528891100073055151954340, 5242880000000000000000001

Offset: 1

Seiichi Manyama, May 08 2021

nonn

Cf. A007429, A023887, A065018, A342471, A344044, A344047, A344061.

a[n_] := DivisorSum[n, DivisorSigma[1 , #]^n &]; Array[a, 19] (* Amiram Eldar, May 08 2021 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (sigma(k)*x)^k/(1-(sigma(k)*x)^k)))

G.f.: Sum_{k >= 1} (sigma(k) * x)^k/(1 - (sigma(k) * x)^k).

If p is prime, a(p) = 1 + (p+1)^p.

cp's OEIS Frontend

A344043 a(n) = n * Sum_{d|n} sigma(d)^3 / d.

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A344047 a(n) = Sum_{d|n} sigma(d)^d.

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A226565 Numbers k such that Sum_{d|k} sigma(d)^3 is a multiple of k.

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A344060 a(n) = Sum_{d|n} sigma(d)^n.

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