A293814 Number of partitions of n into distinct nontrivial divisors of n.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 18, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 13, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 10, 0, 0, 0, 1

Offset: 0

Author

Ilya Gutkovskiy, Oct 16 2017

Keywords

• nonn

Examples

			a(24) = 2 because 24 has 8 divisors {1, 2, 3, 4, 6, 8, 12, 24} among which 6 are nontrivial divisors {2, 3, 4, 6, 8, 12} therefore we have [12, 8, 4] and [12, 6, 4, 2].
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16384
• Index entries for sequences related to partitions

Crossrefs

Cf. A027750, A033630, A070824, A065205, A211111, A293813.

Programs

Maple

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i-1))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 16 2017

Mathematica

Table[d = Divisors[n]; Coefficient[Series[Product[1 + Boole[d[[k]] != 1 && d[[k]] != n] x^d[[k]], {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 100}]

PARI

A293814(n) = { if(!n,return(1)); my(p=1); fordiv(n,d,if(d>1&&dAntti Karttunen, Dec 22 2017

Scheme

;; Implements a simple backtracking algorithm:
(define (A293814 n) (if (<= n 1) (- 1 n) (let ((s (list 0))) (let fork ((r n) (divs (cdr (proper-divisors n)))) (cond ((zero? r) (set-car! s (+ 1 (car s)))) ((or (null? divs) (> (car divs) r)) #f) (else (begin (fork (- r (car divs)) (cdr divs)) (fork r (cdr divs)))))) (car s))))
(define (proper-divisors n) (reverse (cdr (reverse (divisors n)))))
(define (divisors n) (let loop ((k n) (divs (list))) (cond ((zero? k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))
;; Antti Karttunen, Dec 22 2017

Formula

a(n) = [x^n] Product_{d|n, 1 < d < n} (1 + x^d).

a(n) = A211111(n) - 1 for n > 1.