A033630 - OEIS

A033630

Number of partitions of n into distinct divisors of n.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 35, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 2, 1, 7, 1, 1, 1, 26, 1, 1, 1, 2, 1, 24, 1, 1, 1, 1, 1, 22, 1, 1, 1, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (1000 terms from T. D. Noe)

Noah Lebowitz-Lockard and Joseph Vandehey, On the number of partitions of a number into distinct divisors, arXiv:2402.08119 [math.NT], 2024. See p. 2.

FORMULA

a(n) = A065205(n) + 1.

a(A005100(n)) = 1; a(A005835(n)) > 1. - Reinhard Zumkeller, Mar 02 2007

a(n) = f(n, n, 1) with f(n, m, k) = if k <= m then f(n, m, k + 1) + f(n, m - k, k + 1)*0^(n mod k) else 0^m. - Reinhard Zumkeller, Dec 11 2009

a(n) = [x^n] Product_{d|n} (1 + x^d). - Ilya Gutkovskiy, Jul 26 2017

a(n) = 1 if n is deficient (A005100) or weird (A006037). a(n) = 2 if n is perfect (A000396). - Alonso del Arte, Sep 24 2017

EXAMPLE

a(12) = 3 because we have the partitions [12], [6, 4, 2], and [6, 3, 2, 1].

MAPLE

with(numtheory): a:=proc(n) local div, g, gser: div:=divisors(n): g:=product(1+x^div[j], j=1..tau(n)): gser:=series(g, x=0, 105): coeff(gser, x^n): end: seq(a(n), n=1..100); # Emeric Deutsch, Mar 30 2006

# second Maple program:

with(numtheory):

a:= proc(n) local b, l; l:= sort([(divisors(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, Feb 05 2014

MATHEMATICA

A033630 = Table[SeriesCoefficient[Series[Times@@((1 + z^#) & /@ Divisors[n]), {z, 0, n}], n ], {n, 512}] (* Wouter Meeussen *)

A033630[n_] := f[n, n, 1]; f[n_, m_, k_] := f[n, m, k] = If[k <= m, f[n, m, k + 1] + f[n, m - k, k + 1] * Boole[Mod[n, k] == 0], Boole[m == 0]]; Array[A033630, 101, 0] (* Jean-François Alcover, Jul 29 2015, after Reinhard Zumkeller *)

PROG

(Haskell)

a033630 0 = 1

a033630 n = p (a027750_row n) n where

p _ 0 = 1

p [] _ = 0

p (d:ds) m = if d > m then 0 else p ds (m - d) + p ds m

-- Reinhard Zumkeller, Feb 23 2014, Apr 04 2012, Oct 27 2011

CROSSREFS

Cf. A018818, A065205.

Cf. A083206. - Reinhard Zumkeller, Jul 19 2010

Cf. A000009, A005153.