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A327642 Number of partitions of n into divisors of n that are at most sqrt(n). 3
1, 1, 1, 1, 3, 1, 4, 1, 5, 4, 6, 1, 19, 1, 8, 6, 25, 1, 37, 1, 36, 8, 12, 1, 169, 6, 14, 10, 64, 1, 247, 1, 81, 12, 18, 8, 1072, 1, 20, 14, 405, 1, 512, 1, 144, 82, 24, 1, 2825, 8, 146, 18, 196, 1, 1000, 12, 743, 20, 30, 1, 19858, 1, 32, 112, 969, 14, 1728, 1, 324, 24, 1105 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n) > n if n is in A058080 Union {0}, and, a(n) <= n if n is in A007964; indeed, a(n) = n only for n = 1. - Bernard Schott, Sep 22 2019

LINKS

David A. Corneth, Table of n, a(n) for n = 0..9999 (first 5001 terms from Robert Israel)

FORMULA

a(n) = [x^n] Product_{d|n, d <= sqrt(n)} 1 / (1 - x^d).

a(p) = 1, where p is prime.

a(p*q) = q+1 if p <= q are primes. - Robert Israel, Sep 22 2019

EXAMPLE

The divisors of 6 are 1, 2, 3, 6 and sqrt(6) = 2.449..., so the possible partitions are 1+1+1+1+1+1 = 1+1+1+1+2 = 1+1+2+2 = 2+2+2; thus a(6) = 4. - Bernard Schott, Sep 22 2019

MAPLE

f:= proc(n) local x, t, S;

    S:= 1;

    for t in numtheory:-divisors(n) do

      if t^2 <= n then

        S:= series(S/(1-x^t), x, n+1);

      fi

    od;

    coeff(S, x, n);

end proc:

map(f, [$0..100]); # Robert Israel, Sep 22 2019

MATHEMATICA

a[n_] := SeriesCoefficient[Product[1/(1 - Boole[d <= Sqrt[n]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 70}]

PROG

(MAGMA) [1] cat [#RestrictedPartitions(n, {d:d in Divisors(n)| d le Sqrt(n)}):n in [1..70]]; // Marius A. Burtea, Sep 20 2019

CROSSREFS

Cf. A018818, A210442, A211110, A293813.

Sequence in context: A259655 A221185 A242746 * A324242 A216543 A152040

Adjacent sequences:  A327639 A327640 A327641 * A327643 A327644 A327645

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Sep 20 2019

STATUS

approved

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Last modified February 19 22:04 EST 2020. Contains 332060 sequences. (Running on oeis4.)