A248732 -id:A248732 - cp's OEIS frontend

A248728 Number of partitions of 3^n.

1, 3, 30, 3010, 18004327, 133978259344888, 233202632378520643600875145, 817400077628568283525440629036885986580578161120, 37560309092871894517794668078727801667246369744545646936224413217138060330481863103169

Offset: 0

Robert G. Wilson v, Oct 12 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..12

Cf. A000041, A068413, A248730, A248732, A248734, A070177, A248729.

[NumberOfPartitions(3^n): n in [0..8]]; // Vincenzo Librandi, Oct 13 2014

Mathematica
```
Table[ PartitionsP[ 3^n], {n, 0, 8}]
```

a(n) = numbpart(3^n) \\ Michel Marcus, Oct 18 2014

a(n) = A000041(3^n). - Michel Marcus, Oct 18 2014

a(n) ~ exp(Pi*sqrt(2*3^(n-1)))/(4*3^(n+1/2)). - Ilya Gutkovskiy, Jan 13 2017

Added a(0)=1 from Vincenzo Librandi, Oct 13 2014

A248730 Number of partitions of 5^n.

Original entry on oeis.org

1, 7, 1958, 3163127352, 1606903190858354689128371, 8630901377559029573671524821295260243701883575513498104067

Offset: 0

Robert G. Wilson v, Oct 12 2014

nonn

Next term is too big to be included.

Alois P. Heinz, Table of n, a(n) for n = 0..8

Cf. A068413, A248728, A248732, A248734, A070177, A248731.

[NumberOfPartitions(5^n): n in [0..6]]; // Vincenzo Librandi, Oct 13 2014

Mathematica
```
Table[ PartitionsP[ 5^n], {n, 0, 6}]
```

vector(8, n, n--; numbpart(5^n)) \\ Michel Marcus, Oct 13 2014

A248733 Number of digits in the decimal expansion of the number of partitions of 6^n.

Original entry on oeis.org

1, 2, 5, 14, 37, 94, 236, 584, 1437, 3529, 8654, 21210, 51966, 127302, 311840, 763864, 1871094, 4583243

Offset: 0

Robert G. Wilson v, Oct 12 2014

Cf. A129490, A248729, A248731, A248735, A077644, A248736, A248732.

[Floor(Log(10,(NumberOfPartitions(6^n))))+1: n in [0..7]]; // Vincenzo Librandi, Oct 13 2014

f[n_] := Floor[ Log[10, PartitionsP[ 6^n]] + 1]; Table[ f@n, {n, 0, 17}]

a(n) = #Str(numbpart(6^n)); \\ Michel Marcus, Oct 16 2014

from sympy import npartitions
from gmpy2 import digits
def A248733(n): return len(digits(npartitions(6**n))) # Chai Wah Wu, Jul 15 2024

A248733 = A055642 o A000041 o A000400. \\ M. F. Hasler, Oct 16 2014

A248734 Number of partitions of 7^n.

Original entry on oeis.org

1, 15, 173525, 175943559810422753, 229866006383458830949778967121025947053151071434926

Offset: 0

Robert G. Wilson v, Oct 12 2014

nonn

Next term is too big to be included.

Cf. A000041, A068413, A248728, A248730, A248732, A070177, A248735.

[NumberOfPartitions(7^n): n in [0..6]]; // Vincenzo Librandi, Oct 13 2014

Mathematica
```
Table[ PartitionsP[ 7^n], {n, 0, 5}]
```

a(n) = numbpart(7^n) \\ Michel Marcus, Oct 18 2014

a(n) = A000041(7^n). - Michel Marcus, Oct 18 2014

A347615 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is the number of partitions of n^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 22, 30, 5, 1, 1, 1, 231, 3010, 231, 7, 1, 1, 1, 8349, 18004327, 1741630, 1958, 11, 1, 1, 1, 1741630, 133978259344888, 365749566870782, 3163127352, 17977, 15, 1, 1, 1, 4351078600, 233202632378520643600875145, 61847822068260244309086870983975, 1606903190858354689128371, 15285151248481, 173525, 22, 1

Offset: 0

Seiichi Manyama, Sep 08 2021

			Square array begins:
  1, 1,   1,       1,               1, ...
  1, 1,   1,       1,               1, ...
  1, 2,   5,      22,             231, ...
  1, 3,  30,    3010,        18004327, ...
  1, 5, 231, 1741630, 365749566870782, ...

Columns k=0..3 give A000012, A000041, A072213, A128854.
Rows n=0+1, 2-10 give A000012, A068413, A248728, A068413(2*n), A248730, A248732, A248734, A068413(3*n), A248728(2*n), A070177.
Main diagonal gives A347607.
Cf. A238016, A347617, A347618, A347621.

PARI
```
T(n, k) = numbpart(n^k);
```

T(n,k) = A000041(n^k).

cp's OEIS Frontend

A248728 Number of partitions of 3^n.

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A248730 Number of partitions of 5^n.

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A248733 Number of digits in the decimal expansion of the number of partitions of 6^n.

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A248734 Number of partitions of 7^n.

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A347615 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is the number of partitions of n^k.

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