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A018818 Number of partitions of n into divisors of n. 28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Reinhard Zumkeller, Dec 11 2009: (Start)

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)

REFERENCES

Douglas Bowman et al., Partitions of n into parts which are divisors of n, American Mathematical Monthly 99:3 (1992), pp. 276-277.

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 1..10000 (the first 1000 terms were computed by T. D. Noe)

Hansraj Gupta, Partitions of n into divisors of m, Indian J. Pure Appl. Math. 6:11 (1975), pp. 1276-1286.

FORMULA

Coefficient of x^n in expansion of 1/Product_{d divides 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) with f(n,m,k) = if k<=m then f(n,m,k+1)+f(n,m-k,k)*0^(n mod k) else 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

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

CROSSREFS

Cf. A002577, A033630, A171565, A211110, A027750.

Cf. A210442.

Cf. A225244.

Sequence in context: A072478 A190014 A100577 * A157019 A067538 A096154

Adjacent sequences:  A018815 A018816 A018817 * A018819 A018820 A018821

KEYWORD

nonn,nice,look

AUTHOR

David W. Wilson

STATUS

approved

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Last modified May 26 05:25 EDT 2017. Contains 287077 sequences.