A068413 -id:A068413 - cp's OEIS frontend

A114165 Numbers n such that p(2n) is prime, where p(n) is the number of partitions of n.

1, 2, 3, 18, 66, 84, 93, 94, 106, 108, 151, 183, 220, 249, 273, 329, 543, 648, 789, 793, 1068, 1251, 1254, 1284, 1366, 1456, 1549, 1584, 1671, 1771, 2059, 2131, 2228, 2331, 2501, 3399, 3729, 4224, 4456, 4646, 4999, 5093, 5540, 6014, 6510, 6736, 7520, 8124

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

2n-th partition number (A000041(2n)) is prime.

Max Alekseyev, Table of n, a(n) for n = 1..2451

Cf. A000041, A046063, A068413, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.

Select[ Range[9137], PrimeQ[ PartitionsP[2# ]] &]

is(n)=isprime(numbpart(2*n)) \\ Charles R Greathouse IV, Feb 17 2017

A067735 Number of partitions of 2^n into distinct parts.

Original entry on oeis.org

1, 1, 2, 6, 32, 390, 16444, 4013544, 11784471548, 1168225267521350, 16816734263788624008200, 276565526698898057002583240473088, 96052644365764024805972019009272150642974291708, 43586702014259316987395017345466711329303914541873541942193666197800

Offset: 0

Henry Bottomley, Mar 11 2002

nonn

Always even for n>1 since the only powers of two which are generalized pentagonal numbers (A001318 - needed to produce odd numbers of partitions into distinct terms) are 2^0 and 2^1. Number of digits of A068413 divided by number of digits of a(n) approaches sqrt(2).

			a(3)=6 since 2^3=8 can be partitioned into 8, 7+1, 6+2, 5+3, 5+2+1, or 4+3+1.

Alois P. Heinz, Table of n, a(n) for n = 0..14
Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

Cf. A000009, A000079, A068413.

Mathematica
```
Table[ PartitionsQ[2^n], {n, 0, 13}]
```

a(n) = A000009(A000079(n)).

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

A248728 Number of partitions of 3^n.

Original entry on oeis.org

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

A129490 Number of digits in the decimal expansion of the number of partitions of 2^n.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 10, 15, 22, 32, 47, 67, 97, 138, 197, 280, 398, 565, 801, 1134, 1607, 2275, 3219, 4555, 6445, 9118, 12898, 18243, 25803, 36494, 51615, 72998, 103238, 146005, 206486, 292020, 412982, 584050, 825975, 1168110, 1651962, 2336232

Offset: 0

Robert G. Wilson v, Apr 11 2007

For the same sequence but for base 10 (A070177): A077644.

Cf. A000041, A068413, A077644, A129491.

f[n_] := Floor[ Log[10, PartitionsP[2^n]] + 1]; Table[ f@n, {n, 0, 42}]

See A000041: (Hardy and Ramanujan) & (Ayoub, p. 197).
a(n) = A055642(A068413(n)).
a(n) =~ 2*A129491(n)/9.

Missing a(0) prepended by Georg Fischer, Nov 06 2023

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

A248732 Number of partitions of 6^n.

Original entry on oeis.org

1, 11, 17977, 15285151248481, 1398703012615213588677365804960180341, 3173477897288016617984809197028065610087051214582584606785402878333070481745149246796102615681

Offset: 0

Robert G. Wilson v, Oct 12 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..7

Cf. A068413, A248728, A248730, A248734, A070177, A248733.

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

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

A248732 = 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

A327483 Triangle read by rows where T(n,k) is the number of integer partitions of 2^n with mean 2^k, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 22, 34, 8, 1, 1, 231, 919, 249, 16, 1, 1, 8349, 112540, 55974, 1906, 32, 1, 1, 1741630, 107608848, 161410965, 4602893, 14905, 64, 1, 1, 4351078600, 1949696350591, 12623411092535, 676491536028, 461346215, 117874, 128, 1

Offset: 0

Gus Wiseman, Sep 13 2019

T(n,k) is the number of partitions of 2^n into 2^(n-k) parts. - Chai Wah Wu, Sep 21 2023

			Triangle begins:
      1
      1       1
      1       2         1
      1       5         4         1
      1      22        34         8       1
      1     231       919       249      16     1
      1    8349    112540     55974    1906    32  1
      1 1741630 107608848 161410965 4602893 14905 64 1
      ...

Alois P. Heinz, Rows n = 0..13, flattened

Row sums are A327484.
Column k = 1 is A068413 (shifted once to the right).
Cf. A067538, A237984, A240850, A327481, A327482.

Table[Length[Select[IntegerPartitions[2^n],Mean[#]==2^k&]],{n,0,5},{k,0,n}]

from sympy.utilities.iterables import partitions
from sympy import npartitions
def A327483_T(n,k):
    if k == 0 or k == n: return 1
    if k == n-1: return 1<Chai Wah Wu, Sep 21 2023

# uses A008284_T
def A327483_T(n,k): return A008284_T(1<Chai Wah Wu, Sep 21 2023

T(n+1,n) = 2^n for n >= 0. - Chai Wah Wu, Sep 14 2019

a(28)-a(35) from Chai Wah Wu, Sep 14 2019

Row n=8 from Alois P. Heinz, Sep 21 2023

A327484 Number of integer partitions of 2^n whose mean is a power of 2.

Original entry on oeis.org

1, 2, 4, 11, 66, 1417, 178803, 275379307, 15254411521973, 108800468645440803267, 964567296140908420613296779144, 219614169629364529542990295052656098001967511, 38626966436500261962963100479469496821891576834974275502742922521

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

Number of partitions of 2^n whose number of parts is a power of 2. - Chai Wah Wu, Sep 21 2023

			The a(0) = 1 through a(3) = 11 partitions:
  (1)  (2)   (4)     (8)
       (11)  (22)    (44)
             (31)    (53)
             (1111)  (62)
                     (71)
                     (2222)
                     (3221)
                     (3311)
                     (4211)
                     (5111)
                     (11111111)

Chai Wah Wu, Table of n, a(n) for n = 0..15 (n = 0..13 from Alois P. Heinz)

Row sums of A327483.

Cf. A067538, A068413, A135342, A237984, A327474, A327481, A327482.

Table[Length[Select[IntegerPartitions[2^n],IntegerQ[Mean[#]]&]],{n,0,5}]

from sympy.utilities.iterables import partitions
def A327484(n): return sum(1 for s,p in partitions(1<Chai Wah Wu, Sep 21 2023

# uses A008284_T
def A327484(n): return sum(A008284_T(1<Chai Wah Wu, Sep 21 2023

a(7) from Chai Wah Wu, Sep 14 2019
a(8)-a(11) from Alois P. Heinz, Sep 21 2023
a(12) from Chai Wah Wu, Sep 21 2023

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

A114165 Numbers n such that p(2n) is prime, where p(n) is the number of partitions of n.

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A067735 Number of partitions of 2^n into distinct parts.

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A248728 Number of partitions of 3^n.

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A129490 Number of digits in the decimal expansion of the number of partitions of 2^n.

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A248730 Number of partitions of 5^n.

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A248732 Number of partitions of 6^n.

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A248734 Number of partitions of 7^n.

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Magma

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A327483 Triangle read by rows where T(n,k) is the number of integer partitions of 2^n with mean 2^k, 0 <= k <= n.

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