OFFSET
1,2
COMMENTS
For 1 <= k <= n, if k|n then add k * d(k), otherwise add d(k).
If p is prime, a(p) = Sum_{k=1..p} d(k) * k^c(p/k) = 2*p + Sum_{k=1..p-1} d(k) = 2*p - 2 + d(p) + Sum_{k=1..p-1} d(k) = 2*p - 2 + Sum_{k=1..p} d(k).
EXAMPLE
a(8) = Sum_{k=1..8} d(k) * k^c(8/k) = d(1)*1^1 + d(2)*2^1 + d(3)*3^0 + d(4)*4^1 + d(5)*5^0 + d(6)*6^0 + d(7)*7^0 + d(8)*8^1 = 1*1 + 2*2 + 2*1 + 3*4 + 2*1 + 4*1 + 2*1 + 4*8 = 59.
MATHEMATICA
Table[Sum[DivisorSigma[0, k] k^(1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 02 2021
STATUS
approved