A175108 a(n) = number of partitions of d(n) into d(k)'s, where the k's are each <= n and distinct, but the d(k)'s need not be distinct. Here d(m) = the number of divisors of m.

1, 1, 2, 3, 3, 5, 4, 9, 6, 11, 5, 37, 6, 21, 22, 33, 7, 88, 8, 117, 36, 37, 9, 533, 12, 47, 48, 200, 10, 898, 11, 306, 68, 69, 70, 2124, 12, 82, 83, 1864, 13, 2298, 14, 612, 613, 109, 15, 14892, 19, 755, 125, 771, 16, 4669, 141, 4810, 142, 143, 17, 120808, 18, 177, 1265

Offset

1,3

Comments

a(n) = the coefficient of x^d(n) in the product: Product_{k=1..n} (1 + x^d(k)).

Links

Table of n, a(n) for n=1..63.

Example

The first 10 terms of the number-of-divisors sequence (A000005) are 1,2,2,3,2,4,2,4,3,4. For a(10), we want the number of partitions of these 10 terms that sum to d(10)=4. We have: d(10)=4, d(8)=4, d(6)=4, d(9)+d(1) = 3+1, d(4)+d(1) = 3+1, d(7)+d(5) = d(7)+d(3) = d(7)+d(2) = d(5)+d(3) = d(5)+d(2) = d(3)+d(2) = 2+2. So there are 11 such sums (including those equal to 4 itself). Therefore a(10)=11.

Maple

A175108 := proc(n) g := 1 ; for k from 1 to n do g := g*(1+x^numtheory[tau](k)) ; g := expand(g) ; end do ; coeftayl(g, x=0, numtheory[tau](n)) ; end proc: seq(A175108(n), n=1..80) ; # R. J. Mathar, Mar 05 2010

Crossrefs

Cf. A000005.

Sequence in context: A265018 A238792 A158745 * A265145 A103310 A046146

Adjacent sequences: A175105 A175106 A175107 * A175109 A175110 A175111

Keyword

nonn

Author

Leroy Quet, Feb 12 2010

Extensions

More terms from R. J. Mathar and Ray Chandler, Mar 05 2010

Status

approved