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A334739
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Number of unordered factorizations of n with 2 different parts > 1.
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2
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0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 5, 0, 1, 1, 3, 0, 3, 0, 5, 1, 1, 1, 6, 0, 1, 1, 5, 0, 3, 0, 3, 3, 1, 0, 8, 0, 3, 1, 3, 0, 5, 1, 5, 1, 1, 0, 6, 0, 1, 3, 6, 1, 3, 0, 3, 1, 3, 0, 10, 0, 1, 3, 3, 1, 3, 0, 8, 2, 1, 0, 6, 1, 1, 1, 5, 0, 6, 1, 3, 1, 1, 1, 10, 0, 3, 3, 6
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OFFSET
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1,12
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COMMENTS
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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).
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LINKS
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FORMULA
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(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.
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EXAMPLE
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a(24) = 5 = #{ (12,2), (6,4), (8,3), (6,2,2), (3,2,2,2) }.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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