A062051 -id:A062051 - cp's OEIS frontend

A131995 Number of partitions of n into powers of 2 or of 3.

1, 1, 2, 3, 5, 6, 9, 11, 16, 20, 26, 32, 42, 50, 62, 74, 92, 108, 131, 153, 184, 213, 251, 288, 339, 387, 448, 511, 589, 666, 761, 857, 976, 1095, 1237, 1384, 1561, 1737, 1946, 2161, 2415, 2672, 2971, 3281, 3640, 4007, 4425, 4860, 5359, 5869, 6446, 7049, 7729, 8428

Offset: 0

Reinhard Zumkeller, Aug 06 2007

			a(10) = #{9+1, 8+2, 8+1+1, 4+4+2, 4+4+1+1, 4+3+3, 4+3+2+1,
4+3+1+1+1, 4+2+2+2, 4+2+2+1+1, 4+2+1+1+1+1, 4+1+1+1+1+1+1, 3+3+3+1,
3+3+2+2, 3+3+2+1+1, 3+3+1+1+1+1, 3+2+2+2+1, 3+2+2+1+1+1,
3+2+1+1+1+1+1, 3+1+1+1+1+1+1+1, 2+2+2+2+2, 2+2+2+2+1+1, 2+2+2+1+1+1+1,
2+2+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 26.

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A018819, A062051, A023893, A000041, A131996.

g:=(1-x)/(product((1-x^(2^k))*(1-x^(3^k)),k=0..10)): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=1..53); # Emeric Deutsch, Aug 26 2007

G.f.: (1-x)/Product_{k>=0} (1-x^(2^k))*(1-x^(3^k)). - Emeric Deutsch, Aug 26 2007

Prepended a(0) = 1, Joerg Arndt and David A. Corneth, Sep 06 2020

A102434 Sum_{k=1..n} {number of partitions of n into powers k^m where 0<=m

Original entry on oeis.org

1, 5, 14, 43, 136, 477, 1733, 6459, 24338, 92413, 352753, 1352127, 5200351, 20058360, 77558825, 300540275, 1166803192, 4537567749, 17672632001, 68923264531, 269128937347, 1052049482004, 4116715363946, 16123801841726

Offset: 1

Marc LeBrun, Jan 08 2005

Equivalently, Sum_{k=2}^n (Number of partitions of n into powers of k) + Number of partitions of n into n 1's; the latter term is C(2n-1,n).

			a(2) = 5; 3 partitions for k=1: 2.1^0, 1.1^1+1.1^0, 2.1^1; and 2 for k=2: 2.2^0, 1.2^1

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A001700, A102430, A102431, A102432, A102433, A018819, A062051.

a(n) = A102433(n) - n + 1 = A102431(n) + C(2n-1,n).

Edited and verified by Franklin T. Adams-Watters, Mar 10 2006

A309677 G.f. A(x) satisfies: A(x) = A(x^3) / (1 - x)^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 18, 24, 30, 42, 54, 66, 87, 108, 129, 162, 195, 228, 279, 330, 381, 456, 531, 606, 711, 816, 921, 1068, 1215, 1362, 1563, 1764, 1965, 2232, 2499, 2766, 3120, 3474, 3828, 4290, 4752, 5214, 5805, 6396, 6987, 7740, 8493, 9246, 10194, 11142

Offset: 0

Ilya Gutkovskiy, Aug 12 2019

nonn

Self-convolution of A062051.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A005704, A062051, A120880, A171238, A309678, A309679.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(3^i)))
    end:
a:= n-> add(b(j, ilog[3](j))*b(n-j, ilog[3](n-j)), j=0..n):
seq(a(n), n=0..52);  # Alois P. Heinz, Aug 12 2019

nmax = 52; A[] = 1; Do[A[x] = A[x^3]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 52; CoefficientList[Series[Product[1/(1 - x^(3^k))^2, {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1/(1 - x^(3^k))^2.

A131996 Number of partitions of n into distinct powers of 2 or of 3.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12

Offset: 1

Reinhard Zumkeller, Aug 06 2007

nonn

a(A081601(n+1)) = n+1 and a(m) < n+1 for m < A081601(n+1).

			a(10) = #{9+1,8+2,4+3+2+1}=3;
a(20) = #{16+4,16+3+1,9+8+3,9+8+2+1}=4;
a(30) = #{27+3,27+2+1,16+9+4+1,16+9+3+2,16+8+4+2,16+8+3+2+1}=6.

R. Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000009, A062051, A106244, A131995.

g:=(product((1+x^(2^k))*(1+x^(3^k)),k=0..10))/(1+x): gser:=series(g,x=0,111): seq(coeff(gser,x,n),n=1..108); # Emeric Deutsch, Aug 26 2007

max = 100; Product[((1 + x^(2^k)) (1 + x^(3^k))), {k, 0, Log[2, max] // Ceiling}]/(1 + x) + O[x]^max // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Sep 30 2016 *)

G.f.: Product_{k>=0} ((1+x^(2^k))(1+x^(3^k)))/(1+x) (offset 0). - Emeric Deutsch, Aug 26 2007

A294298 Sum of products of terms in all partitions of 3*n into powers of 3.

Original entry on oeis.org

1, 4, 13, 49, 157, 481, 1534, 4693, 14170, 43357, 130918, 393601, 1188454, 3573013, 10726690, 32248957, 96815758, 290516161, 872169223, 2617128409, 7852005967, 23561605318, 70690403371, 212076797530, 636280680100, 1908892327810, 5726727270940, 17180634420931

Offset: 0

Seiichi Manyama, Oct 27 2017

nonn

			n | partitions of 3*n into powers of 3                          | a(n)
----------------------------------------------------------------------------------
1 | 3  , 1+1+1                                                  | 3+1        =  4.
2 | 3+3, 3+1+1+1, 1+1+1+1+1+1                                   | 9+3+1      = 13.
3 | 9  , 3+3+3  , 3+3+1+1+1  , 3+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1 | 9+27+9+3+1 = 49.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A005704, A062051, A289842.

b:= proc(n, i, p) option remember; `if`(n=0, p,
     `if`(i<1, 0, add(b(n-j*i, i/3, p*i^j), j=0..n/i)))
    end:
a:= n-> (t-> b(t, 3^ilog[3](t), 1))(3*n):
seq(a(n), n=0..33);  # Alois P. Heinz, Oct 27 2017

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p, If[i < 1, 0, Sum[b[n - j i, i/3, p i^j], {j, 0, n/i}]]];
a[n_] := b[3n, 3^Floor@Log[3, 3n], 1];
a /@ Range[0, 33] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

a(n) = [x^(3*n)] Product_{k>=0} 1/(1 - 3^k*x^(3^k)). - Ilya Gutkovskiy, Sep 10 2018

a(n) ~ c * 3^n, where c = 2.2530906593645919365992433370928351696108819534655299832797806149219665... - Vaclav Kotesovec, Jun 18 2019

A355915 Number of ways to write n as a sum of numbers of the form 2^r * 3^s, where r and s are >= 0, and no summand divides another.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 1

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Sep 21 2022

nonn

It is a theorem of Erdos [Erdős] that this representation is always possible.
Without the divisibility constraint the answer is A062051.
See A356792 for when k first appears.

			Illustration of initial terms:
1 = 2^0
2 = 2^1
3 = 3^1
4 = 2^2
5 = 2+3
6 = 2*3
7 = 2^2+3
8 = 2^3
9 = 3^2
10 = 2^2 + 2*3
11 = 2+3^2 = 2^3+3 (this is the first time there are 2 solutions)
12 = 2^2*3
13 = 2^2+3^2
14 = 2^3+2*3
...

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Michael S. Branicky, Python Program
William Lowell Putman Mathematical Competition, Number 66, 2005, Problem A-1.

Cf. A062051, A356792.

Python
```
# see linked program
```

More than the usual number of terms are shown, to distinguish this from similar sequences.

A112582 Number of partitions of n into distinct 5-smooth parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 9, 11, 12, 14, 18, 19, 21, 26, 28, 32, 37, 39, 44, 51, 56, 61, 70, 76, 83, 96, 102, 111, 126, 135, 148, 165, 176, 191, 212, 228, 245, 270, 289, 310, 344, 365, 390, 428, 454, 489, 531, 563, 602, 654, 697, 740, 800, 848, 902, 977, 1031, 1093

Offset: 1

Reinhard Zumkeller, Sep 14 2005

nonn

Eric Weisstein's World of Mathematics, Smooth Number
Eric Weisstein's World of Mathematics, Partition Function Q

Cf. A000009, A051037, A062051, A112581.

A272344 Positive integers n where the number of parts function on the set of 3-ary partitions of n is equidistributed mod 3.

Original entry on oeis.org

6, 7, 8, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 33, 34, 35, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 87, 88, 89, 96, 97, 98, 99, 100, 101

Offset: 1

Tom Edgar, Apr 26 2016

nonn

An integer n is in the list if and only if n_i=2 for some index i>0 where n = Sum_{i>=0}n_i3^i is the base 3 representation of n.

Appears to be the complement of A096304.

			There are three 3-ary partitions of 6: one has 2 parts (3+3), one has 4 parts (3+1+1+1), and one has 6 parts (1+1+1+1+1+1); thus, modulo 3, the number of parts function is equidistributed mod 3 and so 6 is a term.
There are five 3-ary partitions of 9 so the number of parts function cannot be equidistributed mod 3. Thus, 9 is not a term.

Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.

Cf. A062051, A096304.

M=[n for n in [1..105] if (2) in n.digits(3)[1:]]

A308558 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 into powers of k > 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 1, 4, 2, 2, 2, 1, 6, 3, 2, 2, 2, 1, 6, 3, 2, 2, 2, 2, 1, 10, 3, 3, 2, 2, 2, 2, 1, 10, 5, 3, 2, 2, 2, 2, 2, 1, 14, 5, 3, 3, 2, 2, 2, 2, 2, 1, 14, 5, 3, 3, 2, 2, 2, 2, 2, 2, 1, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 20, 7, 4, 3, 3, 2

Offset: 1

Gus Wiseman, Jun 07 2019

			Triangle begins:
  1
  1  2
  1  2  2
  1  4  2  2
  1  4  2  2  2
  1  6  3  2  2  2
  1  6  3  2  2  2  2
  1 10  3  3  2  2  2  2
  1 10  5  3  2  2  2  2  2
  1 14  5  3  3  2  2  2  2  2
  1 14  5  3  3  2  2  2  2  2  2
  1 20  7  4  3  3  2  2  2  2  2  2
  1 20  7  4  3  3  2  2  2  2  2  2  2
Row n = 6 counts the following partitions:
  (111111)  (42)      (33)      (411)     (51)      (6)
            (222)     (3111)    (111111)  (111111)  (111111)
            (411)     (111111)
            (2211)
            (21111)
            (111111)

Same as A102430 except for the k = 1 column.
Row sums are A102431(n) + 1.
Column k = 2 is A018819.
Column k = 3 is A062051.
Cf. A000961, A001597, A007916, A008284, A023894, A052410, A101417, A112344, A323053, A323054.

Table[If[k==1,1,Length[Select[IntegerPartitions[n],And@@(IntegerQ[Log[k,#]]&/@#)&]]],{n,10},{k,n}]

A339279 Number of partitions of 3*n into powers of 3 where every part appears at least 2 times.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 13, 15, 18, 22, 25, 29, 34, 38, 43, 50, 55, 62, 70, 77, 85, 95, 103, 113, 126, 136, 149, 164, 177, 192, 210, 225, 243, 265, 283, 305, 330, 352, 377, 406, 431, 460, 494, 523, 557, 595, 629, 667, 710, 748, 791, 841, 884, 934, 989, 1039, 1094, 1156

Offset: 0

Ilya Gutkovskiy, Nov 29 2020

nonn

			a(3) = 3 because we have [3, 3, 3], [3, 3, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000244, A005704, A007690, A062051, A339277.

nmax = 60; CoefficientList[Series[(1/(1 - x^2)) Product[1/(1 - x^(3^k)), {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x]
A005704[0] = 1; A005704[n_] := A005704[n] = A005704[n - 1] + A005704[Floor[n/3]]; a[n_] := Sum[(-1)^(n - k) A005704[k], {k, 0, n}]; Table[a[n], {n, 0, 60}]

G.f.: (1/(1 - x^2)) * Product_{k>=0} 1/(1 - x^(3^k)).
G.f.: (1/(1 - x)) * Product_{k>=0} (1 + x^(2*3^k)/(1 - x^(3^k))).
a(n) = [x^(3*n)] Product_{k>=0} (1 + x^(2*3^k)/(1 - x^(3^k))).
a(n) = Sum_{k=0..n} (-1)^(n-k) * A005704(k).

cp's OEIS Frontend

A131995 Number of partitions of n into powers of 2 or of 3.

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A102434 Sum_{k=1..n} {number of partitions of n into powers k^m where 0<=m

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A309677 G.f. A(x) satisfies: A(x) = A(x^3) / (1 - x)^2.

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Maple

Mathematica

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A131996 Number of partitions of n into distinct powers of 2 or of 3.

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Maple

Mathematica

Formula

A294298 Sum of products of terms in all partitions of 3*n into powers of 3.

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Maple

Mathematica

Formula

A355915 Number of ways to write n as a sum of numbers of the form 2^r * 3^s, where r and s are >= 0, and no summand divides another.

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Python

Extensions

A112582 Number of partitions of n into distinct 5-smooth parts.

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A272344 Positive integers n where the number of parts function on the set of 3-ary partitions of n is equidistributed mod 3.

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