A327642 -id:A327642 - cp's OEIS frontend

A309837 Where records occur in A327642.

0, 4, 6, 8, 10, 12, 16, 18, 24, 30, 36, 48, 60, 72, 84, 96, 108, 120, 144, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 6720, 7560, 9240, 10080, 12600, 13860

Offset: 1

David A. Corneth, Sep 22 2019

nonn

Numbers where the number of partitions of n into divisors of n that are at most sqrt(n) increases to a record.

Is k unbounded where k | a(n) for all n > m for some m. For example, does 2 | a(n) for all n > 1? Does 60 | a(n) for all n > 27?

			There are 1072 partitions of 36 into divisors <= sqrt(36) of 36, i.e. there are 1072 partitions of 36 into parts 1, 2, 3 and 6. For all k < 36, this number of partitions is < 1073 so 36 is in the sequence.

Cf. A327642.

A357311 Number of partitions of n into divisors of n that are smaller than sqrt(n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 4, 1, 5, 1, 6, 1, 19, 1, 8, 6, 9, 1, 37, 1, 36, 8, 12, 1, 169, 1, 14, 10, 64, 1, 247, 1, 81, 12, 18, 8, 478, 1, 20, 14, 405, 1, 512, 1, 144, 82, 24, 1, 2825, 1, 146, 18, 196, 1, 1000, 12, 743, 20, 30, 1, 19858, 1, 32, 112, 289, 14, 1728, 1, 324, 24, 1105

Offset: 0

Ilya Gutkovskiy, Sep 23 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A018818, A210442, A293813, A327642, A357312.

a:= proc(n) option remember; uses numtheory; local b, l;
      l:= sort([select(x-> is(xm, 0, b(m-l[i], i))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Sep 23 2022

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

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

A327766 Number of compositions (ordered partitions) of n into divisors of n that are at most sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 13, 1, 34, 19, 89, 1, 927, 1, 610, 189, 4930, 1, 35890, 1, 46754, 1873, 28657, 1, 3919944, 571, 196418, 18560, 4205249, 1, 110187694, 1, 39882198, 183916, 9227465, 9496, 14484956252, 1, 63245986, 1822473, 11969319436, 1, 141930520462, 1, 34020543362, 339200673

Offset: 0

Ilya Gutkovskiy, Sep 24 2019

nonn

Cf. A100346, A294137, A294138, A327642.

a[n_] := SeriesCoefficient[1/(1 - Sum[Boole[d <= Sqrt[n]] x^d, {d, Divisors[n]}]), {x, 0, n}]; Table[a[n], {n, 0, 45}]

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

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

cp's OEIS Frontend

A309837 Where records occur in A327642.

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A357311 Number of partitions of n into divisors of n that are smaller than sqrt(n).

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A327766 Number of compositions (ordered partitions) of n into divisors of n that are at most sqrt(n).

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Author

Keywords

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Programs

Mathematica

Formula