OFFSET
1,2
COMMENTS
Let sigma_k(n) represent the sum of the k-th powers of the divisors of n.
Then a(n) = Sum_{k=1..n} sigma_k(k), the partial sums of sigma_k(k) for k from 1 to n.
Partial sums of A023887.
LINKS
Jens Kruse Andersen, Table of n, a(n) for n = 1..100
FORMULA
a(n) = Sum_{k=1..n} sigma_k(k).
a(1) = 1. a(n) = a(n-1) + sigma_n(n), for n > 1. - Jens Kruse Andersen, Jul 29 2014
a(n) = n + Sum_{d=2..n} (d^(d*(floor(n/d)+1))-d^d)/(d^d-1). - Chayim Lowen, Aug 04 2015
EXAMPLE
a(1) = 1 because sigma_1(1) = sigma(1) = 1.
a(2) = 6: sigma_1(1) + sigma_2(2) = 1 + (1^2 + 2^2) = 6.
a(3) = 34: sigma_1(1) + sigma_2(2) + sigma_3(3) = 6 + (1^3 + 3^3) = 34.
a(4) = 307: sigma_1(1) + ... + sigma_4(4) = 34 + (1^4 + 2^4 + 4^4) = 307.
MAPLE
B:= [seq(numtheory:-sigma[n](n), n=1..100)]:
seq(add(B[i], i=1..n), n=1..100); # Robert Israel, Jul 28 2014
MATHEMATICA
Table[Sum[DivisorSigma[k, k], {k, n}], {n, 20}]
Accumulate[Table[DivisorSigma[n, n], {n, 20}]] (* Harvey P. Dale, Apr 10 2018 *)
PROG
(PARI)
a(n) = sum(i=1, n, sigma(i, i))
vector(50, n, a(n)) \\ Derek Orr, Jul 27 2014
(Magma) [&+[DivisorSigma(i, i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Jul 29 2014
(Magma) [n eq 1 select 1 else Self(n-1)+ DivisorSigma(n, n): n in [1..20]]; // Vincenzo Librandi, Aug 05 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 22 2014
STATUS
approved