OFFSET
1,2
COMMENTS
From Reinhard Zumkeller, Dec 11 2009: (Start)
From Matthew Crawford, Jan 19 2021: (Start)
For a prime p, a(p^3) = (p^3 + p^2 + 2*p + 4)/2;
For distinct primes p and q, a(p*q) = (p+1)*(q+1)/2 + 2. (End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Douglas Bowman et al., Problem 6640: Partitions of n into parts which are divisors of n, American Mathematical Monthly, 99(3) (1992), 276-277.
Hansraj Gupta, Partitions of n into divisors of m, Indian J. Pure Appl. Math., 6(11) (1975), 1276-1286.
Hansraj Gupta, Partitions of n into divisors of m, Indian J. Pure Appl. Math., 6(11) (1975), 1276-1286.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
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. 1.
Rémy Sigrist, Colored logarithmic scatterplot of the first 10000 terms (where the color is function of A000005(n)).
FORMULA
Coefficient of x^n in the expansion of 1/Product_{d|n} (1-x^d). - Vladeta Jovovic, Sep 28 2002
a(n) = 2 iff n is prime. - Juhani Heino, Aug 27 2009
a(n) = f(n,n,1), where f(n,m,k) = f(n,m,k+1) + f(n,m-k,k)*0^(n mod k) if k <= m, otherwise 0^m. - Reinhard Zumkeller, Dec 11 2009
Paul Erdős, Andrew M. Odlyzko, and the Editors of the AMM give bounds; see Bowman et al. - Charles R Greathouse IV, Dec 04 2012
EXAMPLE
The a(6) = 8 representations of 6 are 6 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1.
MAPLE
A018818 := proc(n)
local a, p, w, el ;
a := 0 ;
for p in combinat[partition](n) do
w := true ;
for el in p do
if modp(n, el) <> 0 then
w := false;
break;
end if;
end do:
if w then
a := a+1 ;
end if;
end do:
a ;
end proc: # R. J. Mathar, Mar 30 2017
MATHEMATICA
Table[d = Divisors[n]; Coefficient[Series[1/Product[1 - x^d[[i]], {i, Length[d]}], {x, 0, n}], x, n], {n, 100}] (* T. D. Noe, Jul 28 2011 *)
PROG
(Haskell)
a018818 n = p (init $ a027750_row n) n + 1 where
p _ 0 = 1
p [] _ = 0
p ks'@(k:ks) m | m < k = 0
| otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Apr 02 2012
(PARI) a(n)=numbpartUsing(n, divisors(n));
numbpartUsing(n, v, mx=#v)=if(n<1, return(n==0)); sum(i=1, mx, numbpartUsing(n-v[i], v, i)) \\ inefficient; Charles R Greathouse IV, Jun 21 2017
(Magma) [#RestrictedPartitions(n, {d:d in Divisors(n)}): n in [1..100]]; // Marius A. Burtea, Jan 02 2019
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved