R, 84 78 bytes

For an arbitrary entry of coprimes, \$a_1, a_2,\ldots\$, Frobenius' amount is given by

a=scan();max((1:(b<-min(a)*max(a)))[-colSums(combn(outer(a,0:b),sum(!!a)))])

Try it online!

since it is always smaller than the product of the extreme \$a_i\$'s. It is then a matter of combining all possible combinations (and more) within that range. Thanks to Cole Beck for suggesting outer with no "*" and colSums rather than `apply(...,2,sum)'.

A faster but longer (by two bytes) version only considers max(a):

a=scan();max((1:(min(a)*(b<-max(a))))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

A slightly shorter version (78 bytes) that most often takes too log or too much space to run on Try it online is

a=scan();max((1:(b<-prod(a)))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

R, 84 78 72 bytes

For an arbitrary entry of coprimes, \$a_1, a_2,\ldots\$, Frobenius' amount is given by

a=scan();max((1:(b<-min(a)*max(a)))[-colSums(combn(outer(a,0:b),sum(!!a)))])

Try it online!

since it is always smaller than the product of the extreme \$a_i\$'s. It is then a matter of combining all possible combinations (and more) within that range. Thanks to Cole Beck for suggesting outer with no "*" and colSums rather than `apply(...,2,sum)'.

A faster but longer (by two bytes) version only considers max(a):

a=scan();max((1:(min(a)*(b<-max(a))))[-colSums(combn(outer(a,0:b),sum(!!a)))])

A slightly shorter version (72 bytes) that most often takes too log or too much space to run on Try it online is

a=scan();max((1:(b<-prod(a)))[-colSums(combn(outer(a,0:b),sum(!!a)))])

R, 84 bytes (78???)

For an arbitrary entry of coprimes, \$a_1, a_2,\ldots\$, Frobenius' amount is given by

a=scan();max((1:(b<-min(a)*max(a)))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

Try it online!

since it is always smaller than the product of the extreme \$a_i\$'s. It is then a matter of combining all possible combinations (and more) within that range. A faster but longer (by two bytes) version only considers max(a):

a=scan();max((1:(min(a)*(b<-max(a))))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

A slightly shorter version (78 bytes) that most often takes too log or too much space to run on Try it online is

a=scan();max((1:(b<-prod(a)))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

R, 84 78 bytes

For an arbitrary entry of coprimes, \$a_1, a_2,\ldots\$, Frobenius' amount is given by

a=scan();max((1:(b<-min(a)*max(a)))[-colSums(combn(outer(a,0:b),sum(!!a)))])

Try it online!

since it is always smaller than the product of the extreme \$a_i\$'s. It is then a matter of combining all possible combinations (and more) within that range. Thanks to Cole Beck for suggesting outer with no "*" and colSums rather than `apply(...,2,sum)'.

A faster but longer (by two bytes) version only considers max(a):

a=scan();max((1:(min(a)*(b<-max(a))))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

A slightly shorter version (78 bytes) that most often takes too log or too much space to run on Try it online is

a=scan();max((1:(b<-prod(a)))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

R, 86 bytes

For an arbitrary entry of coprimes, \$a_1, a_2,\ldots\$, Frobenius' amount is given by

function(a)max((1:(b<-min(a)*max(a)))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

Try it online!

since it is always smaller than the product of the extreme \$a_i\$'s. It is then a matter of combining all possible combinations within that range.

A slightly shorter version (80 bytes) that most often takes too log to run on Try it online is

function(a)max((1:(b<-prod(a)))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

R, 84 bytes (78???)

For an arbitrary entry of coprimes, \$a_1, a_2,\ldots\$, Frobenius' amount is given by

a=scan();max((1:(b<-min(a)*max(a)))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

Try it online!

since it is always smaller than the product of the extreme \$a_i\$'s. It is then a matter of combining all possible combinations (and more) within that range. A faster but longer (by two bytes) version only considers max(a):

a=scan();max((1:(min(a)*(b<-max(a))))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])

A slightly shorter version (78 bytes) that most often takes too log or too much space to run on Try it online is

a=scan();max((1:(b<-prod(a)))[-apply(combn(outer(a,0:b,"*"),sum(!!a)),2,sum)])