OFFSET
1,2
COMMENTS
For prime p, a(p)=3.
a(2^k) = 2^(k+1)-1.
For integers of the form n = p_1*p_2*...*p_k we have a(n) = A007047(k).
The value of a(n) depends only on the exponents in the prime factorization of n.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..20000
FORMULA
Dirichlet g.f.: zeta(s)^2*A(s) where A(s) is the Dirichlet g.f. for A074206. - Geoffrey Critzer, May 23 2018
Sum_{k=1..n} a(k) ~ -4*n^r / (r*Zeta'(r)), where r = A107311 = 1.728647238998183618135103... is the root of the equation zeta(r) = 2. - Vaclav Kotesovec, Jan 31 2019
a(n) = 4*A002033(n-1) - 1 for n > 1. - Geoffrey Critzer, Aug 19 2020
EXAMPLE
a(10) = 11 because we have: {1}, {2}, {5}, {10}, {1|2}, {1|5}, {1|10}, {2|10}, {5|10}, {1|2|10}, {1|5|10}.
MAPLE
with(numtheory):
b:= proc(n) option remember: 1+ `if`(n=1, 0,
add(b(d), d=divisors(n) minus {n}))
end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100); # Alois P. Heinz, Jun 04 2015
MATHEMATICA
Table[Total[Table[Length[Select[Subsets[Divisors[n], {k}], Apply[And, Map[Apply[Divisible, #] &, Partition[Reverse[#], 2, 1]]] &]], {k, 1, PrimeOmega[n] + 1}]], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jun 04 2015
STATUS
approved