A000400 -id:A000400 - cp's OEIS frontend

A248732 Number of partitions of 6^n.

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

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

A013607 a(n) = 6^n-prevprime(6^n).

Original entry on oeis.org

1, 5, 5, 5, 17, 7, 17, 7, 7, 7, 59, 19, 17, 13, 7, 19, 49, 13, 19, 7, 23, 35, 19, 89, 17, 119, 29, 55, 5, 19, 5, 59, 85, 5, 17, 35, 59, 83, 49, 115, 19, 23, 43, 109, 103, 7, 23, 19, 7, 245, 43, 13, 5, 55, 89, 83, 143, 53, 53, 383, 7, 55, 113, 37, 5, 23, 143, 115

Offset: 1

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..1000

Cf. A000400 (6^n), A104091 (largest prime <= 6^n).

Maple
```
seq(6^i-prevprime(6^i),i=1..100);
```

pp[n_]:=Module[{n6=6^n},n6-NextPrime[n6,-1]]; Array[pp,70] (* Harvey P. Dale, May 04 2011 *)
#-NextPrime[#,-1]&/@(6^Range[70]) (* Harvey P. Dale, May 28 2023 *)

a(n) = A000400(n) - A104091(n). - Michel Marcus, Jan 13 2020

A013612 Triangle of coefficients in expansion of (1+5x)^n.

Original entry on oeis.org

1, 1, 5, 1, 10, 25, 1, 15, 75, 125, 1, 20, 150, 500, 625, 1, 25, 250, 1250, 3125, 3125, 1, 30, 375, 2500, 9375, 18750, 15625, 1, 35, 525, 4375, 21875, 65625, 109375, 78125, 1, 40, 700, 7000, 43750, 175000, 437500, 625000, 390625, 1, 45, 900, 10500, 78750, 393750, 1312500, 2812500, 3515625, 1953125

Offset: 0

N. J. A. Sloane

Mirror image of A038243. - Zerinvary Lajos, Nov 25 2007

T(n,k) equals the number of n-length words on {0,1,...,5} having n-k zeros. - Milan Janjic, Jul 24 2015

Alois P. Heinz, Rows n = 0..140, flattened

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+5*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 10 2014

row[n_] := CoefficientList[(1 + 5x)^n, x]; Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Feb 13 2016 *)

G.f.: 1 / [1 - x(1+5y)].

T(n,k) = 5^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k) *C(n,i) *4^(n-i). Row sums are 6^n = A000400(n). - Mircea Merca, Apr 28 2012

A014830 a(1)=1; for n > 1, a(n) = 7*a(n-1) + n.

Original entry on oeis.org

1, 9, 66, 466, 3267, 22875, 160132, 1120932, 7846533, 54925741, 384480198, 2691361398, 18839529799, 131876708607, 923136960264, 6461958721864, 45233711053065, 316635977371473, 2216451841600330, 15515162891202330, 108606140238416331, 760242981668914339, 5321700871682400396

Offset: 1

N. J. A. Sloane

			For n=5, a(5) = 1*15 + 6*20 + 6^2*15 + 6^3*6 + 6^4*1 = 3267. - _Bruno Berselli_, Nov 13 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 18.
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Row n=7 of A126885.

Cf. A000400, A104712.

a:=n->sum((7^(n-j)-1)/6,j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 15 2007

a[1] = 1; a[n_] := 7*a[n-1]+n; Table[a[n], {n, 10}] (* Zak Seidov, Feb 06 2011 *)
LinearRecurrence[{9, -15, 7}, {1, 9, 66}, 30] (* Harvey P. Dale, Jul 22 2013 *)

Vec(x/((1 - x)^2*(1 - 7*x)) + O(x^25)) \\ Colin Barker, Jun 03 2020

a(n) = (7^(n+1) - 6*n - 7)/36. - Rolf Pleisch, Oct 19 2010
a(1)=1, a(2)=9, a(3)=66; for n > 3, a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3). - Harvey P. Dale, Jul 22 2013
a(n) = Sum_{i=0..n-1} 6^i*binomial(n+1,n-1-i). - Bruno Berselli, Nov 13 2015
G.f.: x/((1 - x)^2*(1 - 7*x)). - Colin Barker, Jun 03 2020
E.g.f.: exp(x)*(7*exp(6*x) - 6*x - 7)/36. - Elmo R. Oliveira, Mar 29 2025

A036292 a(n) = n*6^n.

Original entry on oeis.org

0, 6, 72, 648, 5184, 38880, 279936, 1959552, 13436928, 90699264, 604661760, 3990767616, 26121388032, 169789022208, 1097098297344, 7052774768640, 45137758519296, 287753210560512, 1828079220031488, 11577835060199424, 73123168801259520, 460675963447934976

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A000400.

[n*6^n: n in [0..30]]; // Vincenzo Librandi, Apr 10 2016

A036292:=n->n*6^n; seq(A036292(n), n=0..50); # Wesley Ivan Hurt, Nov 09 2013

Table[n*6^n, {n,0,50}] (* Wesley Ivan Hurt, Nov 08 2013 *)
LinearRecurrence[{12, -36}, {0, 6}, 25] (* Vincenzo Librandi, Apr 10 2016 *)

a(n) = n*6^n; \\ Michel Marcus, Nov 09 2013

a(n) = n*A000400(n). - Michel Marcus, Nov 09 2013
From Vincenzo Librandi, Apr 10 2016: (Start)
G.f.: 6*x/(1-6*x)^2.
a(n) = 12*a(n-1) - 36*a(n-2) for n > 1. (End)
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(6/5).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(7/6). (End)
E.g.f.: 6*x*exp(6*x). - Elmo R. Oliveira, Sep 09 2024

A165424 a(1) = 1, a(2) = 6, a(n) = product of the previous terms for n >= 3.

Original entry on oeis.org

1, 6, 6, 36, 1296, 1679616, 2821109907456, 7958661109946400884391936, 63340286662973277706162286946811886609896461828096

Offset: 1

Jaroslav Krizek, Sep 17 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..13

See A173501 for another version. - N. J. A. Sloane, Feb 23 2010

a[1]:= 1; a[2]:= 6; a[n_]:= Product[a[j], {j,1,n-1}]; Table[a[n],{n,1, 12}] (* G. C. Greubel, Oct 19 2018 *)

{a(n) = if(n==1, 1, if(n==2, 6, prod(j=1,n-1, a(j))))};
for(n=1,10, print1(a(n), ", ")) \\ G. C. Greubel, Oct 19 2018

a(1) = 1, a(2) = 6, a(n) = Product_{i=1..n-1} a(i), n >= 3.
a(1) = 1, a(2) = 6, a(n) = A000400(2^(n-3)) = 6^(2^(n-3)), n >= 3.
a(1) = 1, a(2) = 6, a(3) = 6, a(n) = (a(n-1))^2, n >= 4.

A173501 a(1) = 2, a(2) = 3, a(n) = 6^(2^(n-3)) for n >= 3.

Original entry on oeis.org

2, 3, 6, 36, 1296, 1679616, 2821109907456, 7958661109946400884391936, 63340286662973277706162286946811886609896461828096

Offset: 1

Giovanni Teofilatto, Feb 20 2010

nonn

Except for first three terms, a(n) = (a(n-1))^2.

Essentially the same as A165424. - R. J. Mathar, Feb 21 2010

G. C. Greubel, Table of n, a(n) for n = 1..13

Cf. A000079, A000400, A165424.

Table[If[n<3, n+1, 6^(2^(n-3))], {n,10}] (* G. C. Greubel, Apr 25 2021 *)

a(n) = if(n<=2, n+1, 6^(2^(n-3))) \\ Felix Fröhlich, Apr 25 2021

[2,3]+[6^(2^(n-3)) for n in (3..10)] # G. C. Greubel, Apr 25 2021

a(n) = Product_{i=1..n-1} a(i) with a(1) = 2 and a(2) = 3.

a(n) = 6^(2^(n-3)) = A000400(A000079(n-3)) for n>2.

Definition corrected by Glenn Tesler, Aug 19 2017

A175457 (Digit sum of 6^n) mod n.

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 1, 4, 0, 6, 1, 9, 10, 12, 3, 6, 4, 0, 6, 12, 18, 19, 3, 18, 15, 21, 18, 6, 19, 27, 6, 30, 27, 33, 30, 18, 24, 21, 6, 11, 3, 27, 24, 21, 27, 42, 21, 9, 15, 21, 12, 15, 4, 9, 52, 50, 27, 55, 48, 3, 24, 39, 36, 60, 10, 27, 33, 3, 36, 42, 66, 54, 33, 57, 63, 51, 75, 36, 51, 39, 45, 60, 1, 27, 15, 21, 27, 15, 48, 0, 69, 84, 27, 51, 39, 54, 94, 66, 45, 42

Offset: 1

N. J. A. Sloane, Dec 03 2010

Cf. A000400, A066002, A067864.

Table[Mod[Total[IntegerDigits[6^n]],n],{n,120}] (* Harvey P. Dale, May 24 2016 *)

Corrected by N. J. A. Sloane, May 23 2016 at the suggestion of Harvey P. Dale.

A202821 Position of 6^n among 3-smooth numbers A003586.

Original entry on oeis.org

1, 5, 14, 26, 43, 64, 89, 119, 153, 191, 233, 279, 330, 385, 444, 507, 575, 646, 722, 802, 886, 975, 1067, 1164, 1266, 1371, 1481, 1595, 1713, 1835, 1961, 2092, 2227, 2366, 2509, 2657, 2809, 2965, 3125, 3289, 3458, 3630, 3807, 3989, 4174, 4364, 4558, 4756

Offset: 0

Zak Seidov, Dec 25 2011

nonn

			a(0) = 1 because A003586(1) = 6^0 = 1.
a(1) = 5 because A003586(5) = 6^1 = 6.
a(2) = 14 because A003586(14) = 6^2 = 36.

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Zak Seidov)

Cf. A000400, A003586, A022330, A022331.

a[n_] := Sum[Floor[Log[3, 6^n/2^i]] + 1, {i, 0, Log2[6^n]}]; Array[a, 50, 0] (* Amiram Eldar, Jul 15 2023 *)

# uses imports/function in A372401
print(list(islice(A372401gen(p=3), 1000))) # Michael S. Branicky, Jun 06 2024

from sympy import integer_log
def A202821(n): return 1+n*(n+1)+sum((m:=3**i).bit_length()+((1<Chai Wah Wu, Oct 22 2024

A003586(a(n)) = 6^n, for n >= 0.

a(n) ~ (log(6))^2/(log(3)*log(4))*n^2 = 2.1079...*n^2.

cp's OEIS Frontend

A248732 Number of partitions of 6^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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Magma

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PARI

Python

Formula

A013607 a(n) = 6^n-prevprime(6^n).

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Maple

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A013612 Triangle of coefficients in expansion of (1+5x)^n.

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A014830 a(1)=1; for n > 1, a(n) = 7*a(n-1) + n.

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Maple

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A036292 a(n) = n*6^n.

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A165424 a(1) = 1, a(2) = 6, a(n) = product of the previous terms for n >= 3.

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A173501 a(1) = 2, a(2) = 3, a(n) = 6^(2^(n-3)) for n >= 3.

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