OFFSET
1,2
COMMENTS
Binomial transform of the sequence d(n) (A000005). - Emeric Deutsch, May 15 2009
Apparently the partial sums of A101509. - R. J. Mathar, May 17 2009
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric M. Schmidt, The probability that the number of points on a complete intersection is squarefree, Rocky Mountain J. Math, Volume 47, Number 8 (2017), 2777-2796.
FORMULA
G.f.: (Sum_{k>=1} (x/(1-x))^k/(1-x^k/(1-x)^k))/(1-x). - Emeric Deutsch, May 15 2009
E.g.f.: exp(x)*Sum_{k>=1} d(k)*x^k/k!. - Ilya Gutkovskiy, Nov 26 2017
a(n) = 2^n*(log(n) + 2*gamma - log(2)) + O(2^n*n^(-1/4)). [Put alpha_n = beta_n = 1/2 in Thm. 4.2 of Schmidt.] - Eric M. Schmidt, Feb 03 2018
a(n) = Sum_{i=1..n} Sum_{j=1..n} binomial(n,i*j). - Ridouane Oudra, Nov 12 2019
MAPLE
with(numtheory): seq(sum(binomial(n, k)*tau(k), k = 1 .. n), n = 1 .. 30); # Emeric Deutsch, May 15 2009
A160399 := proc(n) local k; add(binomial(n, k)*numtheory[tau](k), k=1..n) ; end: seq(A160399(n), n=1..40) ; # R. J. Mathar, May 17 2009
MATHEMATICA
a[n_] := Sum[Binomial[n, k]*DivisorSigma[0, k], {k, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 25 2017 *)
PROG
(PARI) a(n) = sum(k=1, n, binomial(n, k)*numdiv(k)); \\ Michel Marcus, Feb 25 2017
(GAP) List([1..10^3], n -> Sum([1..n], k -> Binomial(n, k) * Number(DivisorsInt(k)))); # Muniru A Asiru, Feb 04 2018
(Magma) [&+[Binomial(n, k)*NumberOfDivisors(k):k in [1..n]]:n in [1..30]]; // Marius A. Burtea, Nov 12 2019
(Magma) [&+[&+[Binomial(n, i*j):j in [1..n]]:i in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, May 12 2009
EXTENSIONS
More terms from Emeric Deutsch and R. J. Mathar, May 15 2009
STATUS
approved