A070177 -id:A070177 - cp's OEIS frontend

A077644 Number of decimal digits of A070177(n).

1, 2, 9, 32, 107, 347, 1108, 3515, 11132, 35219, 111391, 352269, 1113996, 3522791, 11140072, 35228031, 111400846, 352280442, 1114008610, 3522804578, 11140086260

Offset: 0

Labos Elemer, Nov 15 2002

			p(10^3) = 24061467864032622473692149727991 has 32 decimal digits, so a(3) = 32.

Richard P. Stanley, Enumerative Combinatorics, Cambridge University Press, April 1997, p. 79.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy] p. 825.
Fredrik Johansson, Efficient implementation of the Hardy-Ramanujan-Rademacher formula, 2012 preprint, to be published in LMS Journal of Computation and Mathematics.
Fredrik Johansson, New partition function record: p(10^20) computed (2014)
Herbert S. Wilf, Lectures on Integer Partitions

Cf. A070177.

f[n_] := Floor[ Log[10, PartitionsP[10^n]] + 1]; Array[f, 13, 0]

a(n)=#Str(numbpart(10^n)) \\ Charles R Greathouse IV, Jul 09 2012

a(n) = (Pi*sqrt(2/3)*sqrt(10)^n-log(48)/2-n*log(10))/log(10) + O(1). - Charles R Greathouse IV, Jul 10 2012

a(0), a(10)-a(12), a(15)=35228031 from Robert G. Wilson v, Jun 08 2010
a(13)-a(19) from Charles R Greathouse IV, Jul 09 2012 based on Johansson 2012
a(20) from Robert G. Wilson v, Mar 02 2014

A114170 Numbers n such that p(10n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

44, 1108, 1302, 1504, 1829, 1847, 2267, 2537, 3060, 3289, 3324, 3997, 4138, 6175, 6505, 7266, 9733, 10177, 11483, 12708, 12881, 13632, 14136, 14414, 15917, 16409, 17614, 19133, 19381, 21966, 22967, 30565, 30744, 31655, 33783

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..607

Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170, A070177.

Select[ Range[20000], PrimeQ[PartitionsP[10# ]] &]

is(n)=isprime(numbpart(10*n)) \\ Charles R Greathouse IV, Feb 17 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

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).

A038601 Prime numbers p such that the number of partitions of p is also a prime.

Original entry on oeis.org

2, 3, 5, 13, 157, 491, 863, 1621, 2633, 5347, 8117, 13513, 35227, 62311, 76367, 84017, 141637, 170537, 189353, 192667, 201821, 216617, 251677, 269257, 288203, 293621, 353807, 366103, 367621, 372023, 441703, 444167, 478571, 518657, 582371, 626333, 780707, 816521

Offset: 1

Jeff Burch

nonn

			5 = (1+1+1+1+1+1,1+1+1+2,1+1+3,1+4,1+2+2,2+3,5), so partition(5) = 7; 5 and 7 are primes.

Hisanori Mishima, Partition Number (n = 0 to 1000): Factorizations.
Hisanori Mishima, Partition Number (n = 1001 to 2000): Factorizations.
Hisanori Mishima, Partition Number (n = 2001 to 3000); Factorizations.
Hisanori Mishima, Partition Number (n = 3001 to 4000): Factorizations.
Hisanori Mishima, Partition Number (n = 4001 to 5000): Factorizations.
Hisanori Mishima, Partition Number (n = 5001 to 6000): Factorizations.
Hisanori Mishima, Partition Number (n = 6001 to 7000): Factorizations.
Hisanori Mishima, Partition Number (n = 7001 to 8000): Factorizations.
Hisanori Mishima, Partition Number (n = 8001 to 9000): Factorizations.
Hisanori Mishima, Partition Number (n = 9001 to 10000): Factorizations.
Hisanori Mishima, Factorization of Partition Number.

Cf. A046063, A000041, A070177.

Do[ If[ PrimeQ[n] && PrimeQ[ PartitionsP[n]], Print[n]], {n, 1, 10^5} ]

More terms from Simon Plouffe
More terms from Robert G. Wilson v, Aug 29 2001
a(17)-a(36) from Jacques Tramu, Jun 26 2005
Corrected by T. D. Noe, Oct 31 2006
Offset changed and a(37)-a(38) from Michael S. Branicky, Jun 24 2025

A069878 Number of partitions of 10^n into distinct parts.

Original entry on oeis.org

1, 10, 444793, 8635565795744155161506, 1122606574548038398976040173670530159089991444775125551802871247408332723840

Offset: 0

Robert G. Wilson v, May 03 2002

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..6

Row 10 of A347621.

Cf. A000009, A011557, A064682, A067735 & A070177.

Mathematica
```
Table[ PartitionsQ[10^n], {n, 0, 4}]
```

a(n) = polcoef(prod(k=1, 10^n, 1+x^k+x*O(x^(10^n))), 10^n); \\ Seiichi Manyama, Sep 10 2021

a(n) = A000009(A011557(n)). - Michel Marcus, Sep 10 2021

cp's OEIS Frontend

A077644 Number of decimal digits of A070177(n).

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A114170 Numbers n such that p(10n) is prime, where p(n) is the number of partitions of n.

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PARI

A248728 Number of partitions of 3^n.

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Magma

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

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Mathematica

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A248730 Number of partitions of 5^n.

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Magma

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A248732 Number of partitions of 6^n.

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Magma

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A248734 Number of partitions of 7^n.

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Magma

Mathematica

PARI

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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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