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A018818
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Number of partitions of n into divisors of n.
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104
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1, 2, 2, 4, 2, 8, 2, 10, 5, 11, 2, 45, 2, 14, 14, 36, 2, 81, 2, 92, 18, 20, 2, 458, 7, 23, 23, 156, 2, 742, 2, 202, 26, 29, 26, 2234, 2, 32, 30, 1370, 2, 1654, 2, 337, 286, 38, 2, 9676, 9, 407, 38, 454, 2, 3132, 38, 3065, 42, 47, 2, 73155, 2, 50, 493, 1828, 44, 5257, 2, 740, 50, 5066
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OFFSET
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1,2
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COMMENTS
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For odd primes p: a(p^2) = p + 2; for n > 1: a(A001248(n)) = A052147(n);
For odd primes p > 3, a(3*p) = 2*p + 4; for n > 2: a(A001748(n)) = A100484(n) + 4. (End)
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)
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LINKS
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FORMULA
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Coefficient of x^n in the expansion of 1/Product_{d|n} (1-x^d). - Vladeta Jovovic, Sep 28 2002
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
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EXAMPLE
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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.
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MAPLE
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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 ;
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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