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A033630 Number of partitions of n into distinct divisors of n.

%I A033630 #58 Feb 17 2024 02:46:06
%S A033630 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,
%T A033630 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,
%U A033630 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
%N A033630 Number of partitions of n into distinct divisors of n.
%H A033630 Alois P. Heinz, <a href="/A033630/b033630.txt">Table of n, a(n) for n = 0..10000</a> (1000 terms from T. D. Noe)
%H A033630 Noah Lebowitz-Lockard and Joseph Vandehey, <a href="https://arxiv.org/abs/2402.08119">On the number of partitions of a number into distinct divisors</a>, arXiv:2402.08119 [math.NT], 2024. See p. 2.
%F A033630 a(n) = A065205(n) + 1.
%F A033630 a(A005100(n)) = 1; a(A005835(n)) > 1. - _Reinhard Zumkeller_, Mar 02 2007
%F A033630 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
%F A033630 a(n) = [x^n] Product_{d|n} (1 + x^d). - _Ilya Gutkovskiy_, Jul 26 2017
%F A033630 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
%e A033630 a(12) = 3 because we have the partitions [12], [6, 4, 2], and [6, 3, 2, 1].
%p A033630 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
%p A033630 # second Maple program:
%p A033630 with(numtheory):
%p A033630 a:= proc(n) local b, l; l:= sort([(divisors(n))[]]):
%p A033630       b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A033630              b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i-1))))
%p A033630           end; forget(b):
%p A033630       b(n, nops(l))
%p A033630     end:
%p A033630 seq(a(n), n=0..100); # _Alois P. Heinz_, Feb 05 2014
%t A033630 A033630 = Table[SeriesCoefficient[Series[Times@@((1 + z^#) & /@ Divisors[n]), {z, 0, n}], n ], {n, 512}] (* _Wouter Meeussen_ *)
%t A033630 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_ *)
%o A033630 (Haskell)
%o A033630 a033630 0 = 1
%o A033630 a033630 n = p (a027750_row n) n where
%o A033630    p _  0 = 1
%o A033630    p [] _ = 0
%o A033630    p (d:ds) m = if d > m then 0 else p ds (m - d) + p ds m
%o A033630 -- _Reinhard Zumkeller_, Feb 23 2014, Apr 04 2012, Oct 27 2011
%Y A033630 Cf. A018818, A065205.
%Y A033630 Cf. A083206. - _Reinhard Zumkeller_, Jul 19 2010
%Y A033630 Cf. A000009, A005153.
%Y A033630 Cf. A211111, A027750.
%Y A033630 Cf. A225245.
%Y A033630 Cf. A000396, A005100, A005835, A006037.
%K A033630 nonn
%O A033630 0,7
%A A033630 _Marc LeBrun_
%E A033630 More terms from _Reinhard Zumkeller_, Apr 21 2003