

A236328


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


2



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

1,2


COMMENTS

Sigma(n,k) is the sum of the kth 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


LINKS

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


FORMULA

a(n) = n + Sum_{dn, d>1} d*(d^n1)/(d1).  Chayim Lowen, Aug 02 2015
a(n) >= n*(n^n+n2)/(n1) for n>1.  Chayim Lowen, Aug 05 2015
a(n) = A065805(n)A000005(n).  Chayim Lowen, Aug 11 2015


EXAMPLE

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


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 *)


PROG

(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)/(d1)))) \\ Michel Marcus, Aug 04 2015


CROSSREFS

Cf. A000203, A001157, A001158, A001159, A001160, A109974.
Cf. A000005, A065805.
Cf. A236329.
Sequence in context: A266474 A238723 A316283 * A284337 A052666 A065789
Adjacent sequences: A236325 A236326 A236327 * A236329 A236330 A236331


KEYWORD

nonn,easy


AUTHOR

Colin Barker, Jan 22 2014


STATUS

approved



