A080333 -id:A080333 - cp's OEIS frontend

A080277 Partial sums of A038712.

1, 4, 5, 12, 13, 16, 17, 32, 33, 36, 37, 44, 45, 48, 49, 80, 81, 84, 85, 92, 93, 96, 97, 112, 113, 116, 117, 124, 125, 128, 129, 192, 193, 196, 197, 204, 205, 208, 209, 224, 225, 228, 229, 236, 237, 240, 241, 272, 273, 276, 277, 284, 285, 288, 289, 304, 305, 308

Offset: 1

N. J. A. Sloane, Mar 19 2003

nonn

			From _Omar E. Pol_, Sep 10 2019: (Start)
Illustration of initial terms:
a(n) is also the total area (or the total number of cells) in first n regions of an infinite diagram of compositions (ordered partitions) of the positive integers, where the length of the n-th horizontal line segment is equal to A001511(n), the length of the n-th vertical line segment is equal to A006519(n), and area of the n-th region is equal to A038712(n), as shown below (first eight regions):
-----------------------------------
n  A038712(n)  a(n)       Diagram
-----------------------------------
.                         _ _ _ _
1      1         1       |_| | | |
2      3         4       |_ _| | |
3      1         5       |_|   | |
4      7        12       |_ _ _| |
5      1        13       |_| |   |
6      3        16       |_ _|   |
7      1        17       |_|     |
8     15        32       |_ _ _ _|
.
The above diagram represents the eight compositions of 4: [1,1,1,1],[2,1,1],[1,2,1],[3,1],[1,1,2],[2,2],[1,3],[4].
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..16383
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, 21st Int. Symp. Graph Drawing, 2013, arXiv:1308.0403 [cs.CG], 2013-2014.
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, Journal of Graph Algorithms and Applications 18(2) (2014), 177-209.
Klaus Brockhaus, Illustration of A038712 and A080277.
B. Dearden, J. Iiams, and J. Metzger, A Function Related to the Rumor Sequence Conjecture, J. Int. Seq. 14 (2011), #11.2.3.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions. [ps file]
Ralf Stephan, Table of generating functions. [pdf file]

Cf. A001787, A038712, A050292, A080333, A120385.

A080278 a(n) = (3^(v_3(n) + 1) - 1)/2, where v_3(n) = highest power of 3 dividing n = A007949(n).

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 40, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 40, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 121, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1

Offset: 1

N. J. A. Sloane, Mar 19 2003

Denominator of the quotient sigma(3*n)/sigma(n). - Labos Elemer, Nov 04 2003
a(n) = b/(3*(c+d)) where b, c, d are the sums of the divisors of 3*n that are congruent respectively to 0, 1 and 2 mod 3. - Michel Lagneau, Nov 05 2012
Sum of powers of 3 dividing n. - Amiram Eldar, Nov 27 2022

			a(6) = 4 because the divisors of 3*6 = 18 are {1, 2, 3, 6, 9, 18} => b = 3 + 6 + 9 + 18 = 36, c = 1, d = 2, hence a(6) = b/(3*(c+d)) = 36/(3*(1+2)) = 36/9 = 4. - _Michel Lagneau_, Nov 05 2012

Amiram Eldar, Table of n, a(n) for n = 1..10000
Klaus Brockhaus, Illustration of A080278 and A080333

Cf. A000203, A001620, A007949, A080333, A088838 (numerator of sigma(3*n)/sigma(n)).

Maple
```
A080278 := n->(3^(A007949(n)+1)-1)/2;
```

Table[Denominator[DivisorSigma[1, 3*n]/DivisorSigma[1, n]], {n, 1, 128}]
a[n_] := (3^(IntegerExponent[n, 3] + 1) - 1)/2; Array[a, 100] (* Amiram Eldar, Nov 27 2022 *)

a(n) = denominator(sigma(3*n)/sigma(n)); \\ Michel Marcus, Dec 15 2019

a(n) = (3^(valuation(n, 3) + 1) - 1)/2; \\ Amiram Eldar, Nov 27 2022

G.f.: Sum_{k>=0} 3^k*x^(3^k)/(1-x^(3^k)). - Ralf Stephan, Jun 15 2003
L.g.f.: -log(Product_{k>=0} (1 - x^(3^k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 15 2018
a(n) = sigma(n)/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(3^e) = (3^(e+1)-1)/2, and a(p^e) = 1 for p != 3.
Dirichlet g.f.: zeta(s) / (1 - 3^(1 - s)).
Sum_{k=1..n} a(k) ~ n*log_3(n) + (1/2 + (gamma - 1)/log(3))*n, where gamma is Euler's constant (A001620). (End)

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 1, 5, 0, 0, 0, 0, 0, 1, 2, 5, 0, 0, 0, 0, 0, 1, 1, 2, 12, 0, 0, 0, 0, 0, 0, 1, 1, 2, 12, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 13, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 13

Offset: 0

Pontus von Brömssen, Sep 04 2020

			Array begins:
  n\k|  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
  ---|---------------------------------------------
   0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   1 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   2 |  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   3 |  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0
   4 |  4  1  1  0  0  0  0  0  0  0  0  0  0  0  0
   5 |  4  1  1  1  0  0  0  0  0  0  0  0  0  0  0
   6 |  5  2  1  1  1  0  0  0  0  0  0  0  0  0  0
   7 |  5  2  1  1  1  1  0  0  0  0  0  0  0  0  0
   8 | 12  2  2  1  1  1  1  0  0  0  0  0  0  0  0
   9 | 12  6  2  1  1  1  1  1  0  0  0  0  0  0  0
  10 | 13  6  2  2  1  1  1  1  1  0  0  0  0  0  0
  11 | 13  6  2  2  1  1  1  1  1  1  0  0  0  0  0
  12 | 16  7  3  2  2  1  1  1  1  1  1  0  0  0  0
  13 | 16  7  3  2  2  1  1  1  1  1  1  1  0  0  0
  14 | 17  7  3  2  2  2  1  1  1  1  1  1  1  0  0
  15 | 17  8  3  3  2  2  1  1  1  1  1  1  1  1  0
  16 | 32  8  8  3  2  2  2  1  1  1  1  1  1  1  1
64 = 2*3^3 + 1*3^2 + 0*3^1 + 1*3^0, so T(64,3) = 2*3*3^2 + 1*2*3^1 + 0*1*3^0 = 60. Alternatively, using the formula T(n,k) = floor(n/k) + k*T(floor(n/k),k), we get T(64,3) = 21 + 3*T(21,3) = 21 + 3*(7 + 3*T(7,3)) = 42 + 9*(2 + 3*T(2,3)) = 60.

Pontus von Brömssen, Antidiagonals n = 0..99, flattened
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, Journal of Graph Algorithms and Applications 18(2) (2014), 177-209.

Cf. A136013 (column k=2), A080277 (every second term of column k=2), A080333 (every third term of column k=3).

Cf. A007824, A055129, A286561.

import sympy
def A333979(n,k):
  d=sympy.ntheory.factor_.digits(n,k)
  return sum(j*d[-j-1]*k**(j-1) for j in range(1,len(d)-1))

# Second program (faster)
def A333979(n,k):
  return n//k+k*A333979(n//k,k) if n>=k else 0

T(n,k) = floor(n/k) + k*T(floor(n/k),k). Proof: With n = Sum_{j>=0} d_j*k^j we have floor(n/k) + k*T(floor(n/k),k) = Sum_{j>=1} (d_j*k^(j-1) + k*d_j*(j-1)*k^(j-2)) = Sum_{j>=1} d_j*j*k^(j-1) = T(n,k).
T(n,k) = T(n-1,k) + A055129(A286561(n,k),k). Proof: Let n = Sum_{j>=0} d_j*k^j and pick v so that d_j = 0 for j < v and d_v > 0 (so v = A286561(n,k)). Then n - 1 = sum_{j>=0} e_j*k^j, where e_j = k - 1 for j < v, e_v = d_v - 1, and e_j = d_j for j > v. We get T(n,k) - T(n-1,k) = Sum_{j>=1} j*(d_j-e_j)*k^(j-1) = v*k^(v-1) - (k-1)*Sum_{1<=jA055129(A286561(n,k),k).
For fixed k, T(n,k) ~ n*log(n)/(k*log(k)). (The proof for k = 2 by Bannister et al. (p. 182) can be adapted to general k.)
T(n,k) = Sum_{j>=0} k^j*floor(n/k**(j+1)).

cp's OEIS Frontend

A080277 Partial sums of A038712.

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A080278 a(n) = (3^(v_3(n) + 1) - 1)/2, where v_3(n) = highest power of 3 dividing n = A007949(n).

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A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).

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Python

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cp's OEIS Frontend

A080277 Partial sums of A038712.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A080278 a(n) = (3^(v_3(n) + 1) - 1)/2, where v_3(n) = highest power of 3 dividing n = A007949(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_j*k^j, then T(n,k) = Sum_{j>=1} d_j*j*k^(j-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Python

Formula

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).