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

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

Offset: 1

Author

Colin Barker, Jan 22 2014

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A000203, A001157, A001158, A001159, A001160, A109974.

Cf. A000005, A065805.

Cf. A236329.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

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