A097356 -id:A097356 - cp's OEIS frontend

A368870 Number of partitions of n into nonprime parts not greater than sqrt(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 37, 37, 40, 40, 44, 44, 48, 48, 52, 52, 56, 56, 61, 61, 65, 65, 70, 70, 75, 75, 80, 80, 85, 85, 91, 91, 96, 96, 351, 351, 378, 378, 411, 411, 441

Offset: 0

Ilya Gutkovskiy, Jan 08 2024

nonn

Cf. A002095, A097356.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(isprime(i), 0, b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n, floor(sqrt(n))):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 13 2024

Table[SeriesCoefficient[Product[1/(1 - Boole[!PrimeQ[k]] x^k), {k, 1, Floor[Sqrt[n]]}], {x, 0, n}], {n, 0, 70}]

A369217 Number of partitions of n into squarefree parts not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 4, 4, 5, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 156, 172, 189, 207, 226, 247, 268, 291, 315, 340, 367, 892, 973, 1064, 1159, 1260, 1368, 1485, 1604, 1735, 1872, 2016, 2169, 2333, 5431, 5897, 6394, 6923, 7486, 8088, 8724

Offset: 0

Ilya Gutkovskiy, Jan 16 2024

nonn

Cf. A073576, A097356.

Table[SeriesCoefficient[Product[1/(1 - Boole[SquareFreeQ[k]] x^k), {k, 1, Floor[Sqrt[n]]}], {x, 0, n}], {n, 0, 55}]

A369339 Number of partitions of n into triangular numbers not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 49, 49, 49, 56, 56, 56, 64, 64, 64, 72, 72, 72, 81, 81, 81, 90, 90, 90, 100, 100, 100, 110, 110, 110, 121, 121, 121, 132, 132, 132, 144, 144, 144, 156, 156

Offset: 0

Ilya Gutkovskiy, Jan 20 2024

nonn

Cf. A007294, A097356.

Table[SeriesCoefficient[Product[1/(1 - x^(k (k + 1)/2)), {k, 1, Floor[(Sqrt[1 + 8 Sqrt[n]] - 1)/2]}], {x, 0, n}], {n, 0, 70}]

A369340 Number of partitions of n into squares not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 108, 110, 112, 115, 118

Offset: 0

Ilya Gutkovskiy, Jan 20 2024

nonn

Cf. A001156, A097356.

Table[SeriesCoefficient[Product[1/(1 - x^(k^2)), {k, 1, Floor[n^(1/4)]}], {x, 0, n}], {n, 0, 85}]

cp's OEIS Frontend

A368870 Number of partitions of n into nonprime parts not greater than sqrt(n).

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A369217 Number of partitions of n into squarefree parts not greater than sqrt(n).

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Mathematica

A369339 Number of partitions of n into triangular numbers not greater than sqrt(n).

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Mathematica

A369340 Number of partitions of n into squares not greater than sqrt(n).

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Mathematica