%I #46 Nov 06 2024 04:11:35

%S 1,8,42,374,3910,57210,960806,19261858,435877581,11123320196,

%T 313842837682,9729290348244,328114698808286,11967567841654606,

%U 469172063576559644,19676848703371278522,878942778254232811954,41661071646298278566886,2088331858752553232964218

%N a(n) = sigma(n,1) + sigma(n,2) + ... + sigma(n,n).

%C Sigma(n,k) is the sum of the k-th powers of the divisors of n.

%C The sequence seems to be strictly increasing. - _Chayim Lowen_, Aug 05 2015.

%C 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

%H Robert Israel, <a href="/A236328/b236328.txt">Table of n, a(n) for n = 1..353</a>

%F a(n) = n + Sum_{d|n, d>1} d*(d^n-1)/(d-1). - _Chayim Lowen_, Aug 02 2015

%F a(n) >= n*(n^n+n-2)/(n-1) for n>1. - _Chayim Lowen_, Aug 05 2015

%F a(n) = A065805(n)-A000005(n). - _Chayim Lowen_, Aug 11 2015

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

%p seq(add(numtheory:-sigma[k](n), k=1..n), n=1..50); # _Robert Israel_, Aug 04 2015

%t Table[Sum[DivisorSigma[i, n], {i, n}], {n, 19}] (* _Michael De Vlieger_, Aug 06 2015 *)

%t 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 *)

%o (PARI) vector(30, n, sum(k=1, n, sigma(n, k)))

%o (PARI) vector(30, n, n + sumdiv(n, d, if (d>1,(d^(n+1)-d)/(d-1)))) \\ _Michel Marcus_, Aug 04 2015

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

%Y Cf. A000005, A065805.

%Y Cf. A236329.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 22 2014