OFFSET
1,2
COMMENTS
A multiset is aperiodic if its multiplicities are relatively prime.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = Sum_{d|n} mu(d) * A034691(n/d).
EXAMPLE
The a(4) = 15 aperiodic multisets of compositions are:
{4}, {31}, {22}, {211}, {13}, {121}, {112}, {1111},
{1,3}, {1,21}, {1,12}, {1,111}, {2,11},
{1,1,2}, {1,1,11}.
Missing from this list are {1,1,1,1}, {2,2}, and {11,11}.
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
d*2^(d-1), d=divisors(j))*b(n-j), j=1..n)/n)
end:
a:= n-> add(mobius(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..35); # Alois P. Heinz, Apr 26 2018
MATHEMATICA
nn=20;
ser=Product[1/(1-x^n)^2^(n-1), {n, nn}]
Table[Sum[MoebiusMu[d]*SeriesCoefficient[ser, {x, 0, n/d}], {d, Divisors[n]}], {n, 1, nn}]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(u=EulerT(vector(n, n, 2^(n-1)))); vector(n, n, sumdiv(n, d, moebius(d)*u[n/d]))} \\ Andrew Howroyd, Sep 15 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 26 2018
STATUS
approved