A065805 -id:A065805 - cp's OEIS frontend

A319194 a(n) = Sum_{k=1..n} sigma_n(k).

1, 6, 38, 373, 4461, 68033, 1202753, 24757484, 574608039, 14925278329, 427729375161, 13424413453317, 457608305315211, 16841852554413561, 665483754539870667, 28101844918556128030, 1262901795439193700478, 60182608193322255156347, 3031285556584399354961535

Offset: 1

Vaclav Kotesovec, Sep 13 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..380

Cf. A065805, A308652.

with(NumberTheory): seq(sum(sigma[n](k), k = 1..n), n = 1..20); # Vaclav Kotesovec, Aug 20 2019

Table[Sum[DivisorSigma[n, k], {k, 1, n}], {n, 1, 20}]

a(n) = sum(k=1, n, sigma(k,n)); \\ Michel Marcus, Sep 13 2018

a(n) = sum(k=1, n, k^n * (n\k)); \\ Daniel Suteu, Nov 10 2018

from math import isqrt
from sympy import bernoulli
def A319194(n): return (((s:=isqrt(n))+1)*((b:=bernoulli(n+1))-bernoulli(n+1,s+1))+sum(k**n*(n+1)*((q:=n//k)+1)-b+bernoulli(n+1,q+1) for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023

a(n) ~ n^n / (1 - exp(-1)).

a(n) = Sum_{k=1..n} k^n * floor(n/k). - Daniel Suteu, Nov 10 2018

A236328 a(n) = sigma(n,1) + sigma(n,2) + ... + sigma(n,n).

Original entry on oeis.org

1, 8, 42, 374, 3910, 57210, 960806, 19261858, 435877581, 11123320196, 313842837682, 9729290348244, 328114698808286, 11967567841654606, 469172063576559644, 19676848703371278522, 878942778254232811954, 41661071646298278566886, 2088331858752553232964218

Offset: 1

Colin Barker, Jan 22 2014

Sigma(n,k) is the sum of the k-th powers of the divisors of n.
The sequence seems to be strictly increasing. - Chayim Lowen, Aug 05 2015.
This is true. Moreover, subsequent ratios a(n+1)/a(n) steadily grow for n>3. The difference of subsequent ratios tends to the limit e = 2.718... The reason is that a(n) roughly behaves like n^n; already the second largest term in the sum is smaller by a factor 2^n (for even n) or by a factor 3^n (for n=6k+3) etc. - M. F. Hasler, Aug 16 2015

			a(4) = sigma(4,1) + sigma(4,2) + sigma(4,3) + sigma(4,4) = 7 + 21 + 73 + 273 = 374.

Robert Israel, Table of n, a(n) for n = 1..353

Cf. A000203, A001157, A001158, A001159, A001160, A109974.
Cf. A000005, A065805.
Cf. A236329.

seq(add(numtheory:-sigma[k](n), k=1..n), n=1..50); # Robert Israel, Aug 04 2015

Table[Sum[DivisorSigma[i, n], {i, n}], {n, 19}] (* Michael De Vlieger, Aug 06 2015 *)
f[n_] := Sum[DivisorSigma[i, n], {i, n}]; (* OR *) f[n_] := Block[{d = Rest@Divisors@n}, n + Total[(d^(n + 1) - d)/(d - 1)]]; (* then *) Array[f, 19] (* Robert G. Wilson v, Aug 06 2015 *)

vector(30, n, sum(k=1, n, sigma(n, k)))

vector(30, n, n + sumdiv(n, d, if (d>1,(d^(n+1)-d)/(d-1)))) \\ Michel Marcus, Aug 04 2015

a(n) = n + Sum_{d|n, d>1} d*(d^n-1)/(d-1). - Chayim Lowen, Aug 02 2015
a(n) >= n*(n^n+n-2)/(n-1) for n>1. - Chayim Lowen, Aug 05 2015
a(n) = A065805(n)-A000005(n). - Chayim Lowen, Aug 11 2015
a(n) ~ n^n. - Vaclav Kotesovec, Aug 04 2025

cp's OEIS Frontend

A319194 a(n) = Sum_{k=1..n} sigma_n(k).

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A236328 a(n) = sigma(n,1) + sigma(n,2) + ... + sigma(n,n).

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