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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 LINKS Robert Israel, Table of n, a(n) for n = 1..353 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 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)/(d-1)))) \\ 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

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Last modified June 4 03:55 EDT 2023. Contains 363118 sequences. (Running on oeis4.)