A005705 -id:A005705 - cp's OEIS frontend

A268443 a(n) = (A005705(n) - A268444(n))/4.

0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 3, 6, 6, 6, 6, 11, 12, 13, 14, 17, 18, 19, 20, 25, 26, 27, 28, 35, 36, 37, 38, 49, 52, 55, 58, 64, 67, 70, 73, 82, 85, 88, 91, 103, 106, 109, 112, 130, 136, 142, 148, 158, 164, 170, 176, 190, 196, 202, 208, 226, 232, 238, 244

Offset: 0

Tom Edgar, Feb 04 2016

nonn

G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.

Cf. A005705, A268444, A268127, A268128.

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[i//4])
    return A[n]
print([(b(n)-prod(x+1 for x in n.digits(4)))/4 for n in [0..63]])

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/4)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*4^i is the base 4 representation of n. Then a(n) = (1/4)*(b(n) - c(n)).

A005704 Number of partitions of 3n into powers of 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 15, 18, 23, 28, 33, 40, 47, 54, 63, 72, 81, 93, 105, 117, 132, 147, 162, 180, 198, 216, 239, 262, 285, 313, 341, 369, 402, 435, 468, 508, 548, 588, 635, 682, 729, 783, 837, 891, 954, 1017, 1080, 1152, 1224, 1296, 1377, 1458, 1539, 1632

Offset: 0

N. J. A. Sloane

nonn

Infinite convolution product of [1,2,3,3,3,3,3,3,3,3] aerated A000244 - 1 times, i.e., [1,2,3,3,3,3,3,3,3,3] * [1,0,0,2,0,0,3,0,0,3] * [1,0,0,0,0,0,0,0,0,2] * ... [Mats Granvik, Gary W. Adamson, Aug 07 2009]

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
G. E. Andrews, Congruence properties of the m-ary partition function, J. Number Theory 3 (1971), 104-110.
G. E. Andrews and J. A. Sellers, Characterizing the number of p-ary partitions modulo a prime p, p. 2.
C. Banderier, H.-K. Hwang, V. Ravelomanana, and V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics, 2012. - From _N. J. A. Sloane_, Dec 23 2012
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, Australasian J. Combin., 30 (2004), 193-196.
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions
D. Krenn, D. Ralaivaosaona, and S. Wagner, The Number of Multi-Base Representations of an Integer, 2014.
D. Krenn, D. Ralaivaosaona, and S. Wagner, Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems, arXiv:1503.08594 [math.NT], 2015. Also in Applicable Analysis and Discrete Mathematics (2015) Vol. 9, Issue 2, 285-312.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]
O. J. Rodseth, Some arithmetical properties of m-ary partitions, Proc. Camb. Phil. Soc. 68 (1970), 447-453.
O. J. Rodseth and J. A. Sellers, On m-ary partition function congruences: A fresh look at a past problem, J. Number Theory 87 (2001), 270-281.
O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.

Cf. A000041, A000123, A005705, A005706, A018819.

Cf. A006996.

Fold[Append[#1, Total[Take[Flatten[Transpose[{#1, #1, #1}]], #2]]] &, {1}, Range[2, 55]] (* Birkas Gyorgy, Apr 18 2011 *)
a[n_] := a[n] = If[n <= 2, n + 1, a[n - 1] + a[Floor[n/3]]]; Array[a, 101, 0] (* T. D. Noe, Apr 18 2011 *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A005704(n): return A005704(n-1)+A005704(n//3) if n else 1 # Chai Wah Wu, Sep 21 2022

a(n) = a(n-1)+a(floor(n/3)).
Coefficient of x^(3*n) in prod(k>=0, 1/(1-x^(3^k))). Also, coefficient of x^n in prod(k>=0, 1/(1-x^(3^k)))/(1-x). - Benoit Cloitre, Nov 28 2002
a(n) mod 3 = binomial(2n, n) mod 3. - Benoit Cloitre, Jan 04 2004
Let T(x) be the g.f., then T(x)=(1-x^3)/(1-x)^2*T(x^3). [Joerg Arndt, May 12 2010]

Formula and more terms from Henry Bottomley, Apr 30 2001

A005706 Number of partitions of 5n into powers of 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 82, 89, 96, 103, 110, 119, 128, 137, 146, 155, 166, 177, 188, 199, 210, 223, 236, 249, 262, 275, 290, 305, 320, 335, 350, 368, 386, 404, 422, 440, 461, 482, 503, 524, 545

Offset: 0

N. J. A. Sloane

Euler transform of [2,0,0,0,1,0,0,0,0,...] with 1's at 5^n. - Michael Somos, Mar 16 2004

Partial sums of number of partitions of n into powers of 5. - Michael Somos, Mar 16 2004

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
C. Banderier, H.-K. Hwang, V. Ravelomanana and V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics, preprint 2012; SIAM J. Discrete Math., 28(1), 342-371, 2014. - _N. J. A. Sloane_, Dec 23 2012
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, Australasian J. Combin., 30 (2004), 193-196.
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]
O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.

Column k=5 of A292477.

Cf. A000041, A000123, A005704, A005705.

a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/5]]; Table[a@ n, {n, 0, 60}] (* Michael De Vlieger, Mar 25 2016 *)

PARI
```
a(n)=if(n<1,n==0,a(n-1)+a(n\5))
```

a(n) = a(n-1) + a([n/5]).

a(n) = [x^(5*n)] Product_{k>=0} 1/(1 - x^(5^k)). - Ilya Gutkovskiy, Jun 05 2017

Formula and more terms from Henry Bottomley, Apr 30 2001

A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 17, 23, 24, 34, 35, 43, 47, 57, 58, 73, 74, 91, 96, 112, 113, 139, 141, 163, 168, 197, 198, 235, 236, 272, 279, 317, 321, 378, 379, 427, 436, 501, 502, 575, 576, 653, 666, 742, 743, 851, 853, 952, 963, 1080, 1081, 1211, 1216, 1361

Offset: 0

Henry Bottomley, Jul 05 2002

nonn

First differs from A322912 at a(12) = 34, A322912(12) = 33.

			a(6)=10 since 6 can be written as 6 (powers of 6), 5+1 (5), 4+1+1 (4 or 2), 3+3 (3), 3+1+1+1 (3), 4+2 (2), 2+2+2 (2), 2+2+1+1 (2), 2+1+1+1+1 (2) and 1+1+1+1+1+1 (powers of anything).
From _Gus Wiseman_, Jan 01 2019: (Start)
The a(1) = 1 through a(8) = 15 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (42)      (331)      (71)
                    (211)   (311)    (51)      (421)      (422)
                    (1111)  (2111)   (222)     (511)      (611)
                            (11111)  (411)     (2221)     (2222)
                                     (2211)    (4111)     (3311)
                                     (3111)    (22111)    (4211)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Cf. A000123, A005704, A005705, A005706, A072721.

Cf. A018819, A023893, A052410, A102430, A322900, A322901, A322902, A322903, A322912.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@DeleteCases[#,1]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)

a(n) = a(n-1) + A072721(n). a(p) = a(p-1)+1 for p prime.

A292477 Square array A(n,k), n >= 0, k >= 2, read by antidiagonals: A(n,k) = [x^(k*n)] Product_{j>=0} 1/(1 - x^(k^j)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 6, 1, 2, 3, 5, 10, 1, 2, 3, 4, 7, 14, 1, 2, 3, 4, 6, 9, 20, 1, 2, 3, 4, 5, 8, 12, 26, 1, 2, 3, 4, 5, 7, 10, 15, 36, 1, 2, 3, 4, 5, 6, 9, 12, 18, 46, 1, 2, 3, 4, 5, 6, 8, 11, 15, 23, 60, 1, 2, 3, 4, 5, 6, 7, 10, 13, 18, 28, 74, 1, 2, 3, 4, 5, 6, 7, 9, 12, 15, 21, 33, 94

Offset: 0

Ilya Gutkovskiy, Sep 17 2017

A(n,k) is the number of partitions of k*n into powers of k.

			Square array begins:
   1,  1,  1,  1,  1,  1, ...
   2,  2,  2,  2,  2,  2, ...
   4,  3,  3,  3,  3,  3, ...
   6,  5,  4,  4,  4,  4, ...
  10,  7,  6,  5,  5,  5, ...
  14,  9,  8,  7,  6,  6, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
George E. Andrews, Aviezri S. Fraenkel, James A. Sellers, Characterizing the number of m-ary partitions modulo m, Amer. Math. Monthly 122:9 (2015), 880-885.
Index entries for related partition-counting sequences

Columns k=2..5 give A000123, A005704, A005705, A005706.
Mirror of A089688 (excluding the first row).
Cf. A145515, A294316.

Table[Function[k, SeriesCoefficient[Product[1/(1 - x^k^i), {i, 0, n}], {x, 0, k n}]][j - n + 2], {j, 0, 12}, {n, 0, j}] // Flatten

A089688 Table T(n,k), n>=0 and k>=1, read by antidiagonals; the k-th row is defined by : partitions of k*n into powers of k (with T(0,k) = 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 2, 1, 1, 10, 5, 3, 2, 1, 1, 14, 7, 4, 3, 2, 1, 1, 20, 9, 6, 4, 3, 2, 1, 1, 26, 12, 8, 5, 4, 3, 2, 1, 1, 36, 15, 10, 7, 5, 4, 3, 2, 1, 1, 46, 18, 12, 9, 6, 5, 4, 3, 2, 1, 1, 60, 23, 15, 11, 8, 6, 5, 4, 3, 2, 1

Offset: 0

Philippe Deléham, Jan 05 2004

			Row k = 1 : 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1, ... (see A000012).
Row k = 2 : 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, ... (see A000123).
Row k = 3 : 1, 2, 3, 5,  7,  9, 12, 15, 18, 23, 28, 33, ... (see A005704).
Row k = 4 : 1, 2, 3, 4,  6,  8, 10, 12, 15, 18, 21, 24, ... (see A005705).
Row k = 5 : 1, 2, 3, 4,  5,  7,  9, 11, 13, 15, 18, 21, ... (see A005706).

Alois P. Heinz, Table of n, a(n) for n = 0..10010

A309678 G.f. A(x) satisfies: A(x) = A(x^4) / (1 - x)^2.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 16, 22, 28, 34, 40, 50, 60, 70, 80, 97, 114, 131, 148, 175, 202, 229, 256, 296, 336, 376, 416, 472, 528, 584, 640, 718, 796, 874, 952, 1058, 1164, 1270, 1376, 1516, 1656, 1796, 1936, 2116, 2296, 2476, 2656, 2886, 3116, 3346, 3576, 3866, 4156, 4446, 4736, 5096

Offset: 0

Ilya Gutkovskiy, Aug 12 2019

nonn

Cf. A005705, A171238, A174026, A309677, A309679.

nmax = 56; A[] = 1; Do[A[x] = A[x^4]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 56; CoefficientList[Series[Product[1/(1 - x^(4^k))^2, {k, 0, Floor[Log[4, nmax]] + 1}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1/(1 - x^(4^k))^2.

A272270 Positive integers n where the number of parts function on the set of 4-ary partitions of n is equidistributed mod 4.

Original entry on oeis.org

12, 13, 14, 15, 28, 29, 30, 31, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 76, 77, 78, 79, 92, 93, 94, 95, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 140, 141, 142, 143

Offset: 1

Tom Edgar, Apr 28 2016

nonn

An integer n is in the list if and only if n_i=3 for some index i>0 where n = Sum_{i>=0}n_i4^i is the base 4 representation of n.

			There are four 4-ary partitions of 12: one has 12 parts (1+1+1+1+1+1+1+1+1+1+1+1), one has 3 parts (4+4+4), one has 9 parts (4+1+1+1+1+1+1+1+1), and one has 6 parts (4+4+1+1+1+1); thus, modulo 4, the number of parts function is equidistributed mod 4 and so 12 is a term.
There are six 4-ary partitions of 16 so the number of parts function cannot be equidistributed mod 4. Thus, 16 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. A005705, A272344.

[n for n in [1..150] if 3 in n.digits(4)[1:]]

cp's OEIS Frontend

A268443 a(n) = (A005705(n) - A268444(n))/4.

Views

Author

Keywords

Links

Crossrefs

Programs

Sage

Formula

A005704 Number of partitions of 3n into powers of 3.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A005706 Number of partitions of 5n into powers of 5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A292477 Square array A(n,k), n >= 0, k >= 2, read by antidiagonals: A(n,k) = [x^(k*n)] Product_{j>=0} 1/(1 - x^(k^j)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A089688 Table T(n,k), n>=0 and k>=1, read by antidiagonals; the k-th row is defined by : partitions of k*n into powers of k (with T(0,k) = 1).

Views

Author

Keywords

Examples

Links

A309678 G.f. A(x) satisfies: A(x) = A(x^4) / (1 - x)^2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A272270 Positive integers n where the number of parts function on the set of 4-ary partitions of n is equidistributed mod 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage