A290474 Number of fractional partitions of n where each element of a partition is a rational number r/s such that r < s, s <= n, and gcd(r,s) = 1.

1, 0, 1, 12, 135, 4477, 100160, 8663934, 485380025, 80730951180, 10180011676356, 4126137351376215, 563950787766342780, 456369006693283278869, 200330760220853808357439, 335435016971402890883460861, 197675615401466868237710861644, 561969529551274362018496511765678

Offset: 0

Author

Joseph Wheat, Aug 03 2017

Keywords

• nonn

Comments

a(n) = (n^2 + 1)^(-1 + Sum_{k=1..n} phi(k)) - f(n) where phi(n) is Euler's totient function, and f(n) is the number of trivial solutions which do not satisfy the equation q_1*x_1 + q_2*x_2 + ... + q_m*x_m = n. Each coefficient is a rational number satisfying the criteria given in the definition, and m = -1 + Sum_{k=1..n} phi(k).

Examples

			For n=3, the number of partitions is equal to the number of nonnegative integer solutions for the equation: (1/2)*x_1 + (1/3)*x_2 + (2/3)*x_3 = 3. The set S of solutions is {[0,1,4], [0,3,3], [0,5,2], [0,7,1], [0,9,0], [2,0,3], [2,2,2], [2,4,1], [2,6,0], [4,1,1], [4,3,0], [6,0,0]}. Therefore, |S| = a(3) = 12.
		

Crossrefs

Cf. A015614, A057562, A154886, A154887, A171978.

Programs

PARI

s(v, n, t) = {if(t==#v, f = n\v[t]; v[t]*f == n, sum(i=0, n\v[t], s(v, n-v[t]*i, t+1)))}
a(n) = {if(n<=2, return(n-1)); my(fractions = List(), q = 0); for(i=2, n, for(j=1, i-1, if(gcd(i, j)==1, listput(fractions, j/i)))); s(fractions, n, 1)} \\ David A. Corneth, Aug 03 2017

Extensions

a(7)-a(17) from Alois P. Heinz, Aug 03 2017