OFFSET
1,12
COMMENTS
a(n) depends only on the prime signature of n. E.g., a(12)=a(75), since 12=2^2*3 and 75=5^2*3 share the same prime signature (2,1).
LINKS
FORMULA
(Joint) D.g.f.: Product_{n>=2} ( 1 + t/(n^s-1) ).
Recursion: a(n) = h_2(n), where h_l(n) * log(n) = Sum_{ d^i | n } Sum_{j=1..l} (-1)^(j+1) * h_{l-j}(n/d^i) * log(d), with h_l(n)=1 if n=1 and l=0 otherwise h_l(n)=0.
EXAMPLE
a(24) = 5 = #{ (12,2), (6,4), (8,3), (6,2,2), (3,2,2,2) }.
PROG
(R)
maxe <- function(n, d) { i=0; while( n%%(d^(i+1))==0 ) { i=i+1 }; i }
uhRec <- function(n, l=1) {
uh = 0
if( n<=0 ) {
return(0)
} else if(n==1) {
return(ifelse(l==0, 1, 0))
} else if(l<=0) {
return(0)
} else if( (n>=2) && (l>=1) ) {
for(d in 2:n) {
m = maxe(n, d)
if(m>=1) for(i in 1:m) for(j in 1:min(i, l)) {
uhj = uhRec( n/d^i, l-j )
uh = uh + log(d)/log(n) * (-1)^(j+1) * choose(i, j) * uhj
}
}
return(round(uh, 3))
}
}
n=100; l=2; sapply(1:n, uhRec, l) # A334739
n=100; l=3; sapply(1:n, uhRec, l) # A334740
CROSSREFS
KEYWORD
nonn
AUTHOR
Jacob Sprittulla, May 09 2020
STATUS
approved