A113999 -id:A113999 - cp's OEIS frontend

A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

1, 3, 5, 11, 17, 39, 65, 139, 261, 531, 1025, 2095, 4097, 8259, 16405, 32907, 65537, 131367, 262145, 524827, 1048645, 2098179, 4194305, 8390831, 16777233, 33558531, 67109125, 134225995, 268435457, 536887863, 1073741825, 2147516555, 4294968325, 8590000131

Offset: 1

Erich Friedman

nonn

Dirichlet convolution of b_n=1 with c_n = 2^(n-1).
Equals row sums of triangle A143425, & inverse Möbius transform (A051731) of [1, 2, 4, 8, ...]. - Gary W. Adamson, Aug 14 2008
Number of constant multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The a(4) = 11 constant multiset partitions:
  (1)(1)(1)(1)
    (11)(11)
    (12)(12)
     (1111)
     (1222)
     (1122)
     (1112)
     (1233)
     (1223)
     (1123)
     (1234)
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005, A002033, A003238, A008480, A018783, A047968, A051731.
Cf. A052409, A055895, A078392, A143425, A215366, A245282, A248906.
Cf. A289508.
Sums of the form Sum_{d|n} q^(d-1): this sequence (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A034729:= func< n | (&+[2^(d-1): d in Divisors(n)]) >;
[A034729(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

seq(add(2^(k-1),k=numtheory:-divisors(n)), n = 1 .. 100); # Robert Israel, Aug 22 2014

Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k),{k,1,30}],{x,0,30}],x]] (* Vaclav Kotesovec, Sep 08 2014 *)

A034729(n) = sumdiv(n,k,2^(k-1)) \\ Michael B. Porter, Mar 11 2010

{a(n)=polcoeff(sum(m=1,n,2^(m-1)*x^m/(1-x^m +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014

{a(n)=local(A=x+x^2);A=sum(m=1,n,x^m*sumdiv(m,d,1/(1 - x^(m/d) +x*O(x^n))^d) );polcoeff(A,n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014

from sympy import divisors
def A034729(n): return sum(1<<(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 15 2022

def A034729(n): return sum(2^(k-1) for k in (1..n) if (k).divides(n))
[A034729(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

G.f.: Sum_{n>0} x^n/(1-2*x^n). - Vladeta Jovovic, Nov 14 2002
a(n) = 1/2 * A055895(n). - Joerg Arndt, Aug 14 2012
G.f.: Sum_{n>=1} 2^(n-1) * x^n / (1 - x^n). - Paul D. Hanna, Aug 21 2014
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1 - x^d)^(n/d). - Paul D. Hanna, Aug 21 2014
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 09 2014
a(n) = Sum_{k in row n of A215366} A008480(k) * A000005(A289508(k)). - Gus Wiseman, Sep 16 2018
a(n) = Sum_{c is a composition of n} A000005(gcd(c)). - Gus Wiseman, Sep 16 2018

A034730 Dirichlet convolution of b_n=1 with c_n=3^(n-1).

Original entry on oeis.org

1, 4, 10, 31, 82, 256, 730, 2218, 6571, 19768, 59050, 177430, 531442, 1595056, 4783060, 14351125, 43046722, 129146980, 387420490, 1162281262, 3486785140, 10460412256, 31381059610, 94143358444, 282429536563, 847289140888, 2541865834900, 7625599080070, 22876792454962

Offset: 1

Erich Friedman

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..2000

Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), this sequence (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A034730:= func< n | (&+[3^(d-1): d in Divisors(n)]) >;
[A034730(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Rest[CoefficientList[Series[Sum[x^k/(1-3*x^k),{k,1,30}],{x,0,30}],x]] (* Vaclav Kotesovec, Sep 09 2014 *)
A034730[n_]:= DivisorSum[n, 3^(#-1) &];
Table[A034730[n], {n,40}] (* G. C. Greubel, Jun 25 2024 *)

{a(n) = sumdiv(n, d, 3^(d-1))} \\ Seiichi Manyama, Jun 26 2019

def A034730(n): return sum(3^(k-1) for k in (1..n) if (k).divides(n))
[A034730(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{n>0} x^n/(1-3*x^n). - Vladeta Jovovic, Nov 14 2002
a(n) ~ 3^(n-2). - Vaclav Kotesovec, Sep 09 2014
a(n) = Sum_{d|n} 3^(d-1). - Seiichi Manyama, Jun 26 2019

A339684 a(n) = Sum_{d|n} 4^(d-1).

Original entry on oeis.org

1, 5, 17, 69, 257, 1045, 4097, 16453, 65553, 262405, 1048577, 4195413, 16777217, 67112965, 268435729, 1073758277, 4294967297, 17179935765, 68719476737, 274878169413, 1099511631889, 4398047559685, 17592186044417, 70368748389461, 281474976710913

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000302, A295505.

Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), this sequence (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A339684:= func< n | (&+[4^(d-1): d in Divisors(n)]) >;
[A339684(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[4^(d - 1), {d, Divisors[n]}], {n, 1, 25}]
nmax = 25; CoefficientList[Series[Sum[x^k/(1 - 4 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 4^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339684(n): return sum(4^(k-1) for k in (1..n) if (k).divides(n))
[A339684(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 4*x^k).
G.f.: Sum_{k>=1} 4^(k-1) * x^k / (1 - x^k).
a(n) ~ 4^(n-1). - Vaclav Kotesovec, Jun 05 2021

A339685 a(n) = Sum_{d|n} 5^(d-1).

Original entry on oeis.org

1, 6, 26, 131, 626, 3156, 15626, 78256, 390651, 1953756, 9765626, 48831406, 244140626, 1220718756, 6103516276, 30517656381, 152587890626, 762939846906, 3814697265626, 19073488282006, 95367431656276, 476837167968756, 2384185791015626, 11920929003987656

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 5 of A308813.
Cf. A000351, A295506.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), this sequence (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A339685:= func< n | (&+[5^(d-1): d in Divisors(n)]) >;
[A339685(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[5^(d - 1), {d, Divisors[n]}], {n, 1, 24}]
nmax = 24; CoefficientList[Series[Sum[x^k/(1 - 5 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 5^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339685(n): return sum(5^(k-1) for k in (1..n) if (k).divides(n))
[A339685(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 5*x^k).
G.f.: Sum_{k>=1} 5^(k-1) * x^k / (1 - x^k).
a(n) ~ 5^(n-1). - Vaclav Kotesovec, Jun 05 2021

A339686 a(n) = Sum_{d|n} 6^(d-1).

Original entry on oeis.org

1, 7, 37, 223, 1297, 7819, 46657, 280159, 1679653, 10078999, 60466177, 362805091, 2176782337, 13060740679, 78364165429, 470185264735, 2821109907457, 16926661132171, 101559956668417, 609359750089711, 3656158440109669, 21936950700844039, 131621703842267137

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 6 of A308813.
Cf. A000400, A320071.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), this sequence (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A339686:= func< n | (&+[6^(d-1): d in Divisors(n)]) >;
[A339686(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[6^(d - 1), {d, Divisors[n]}], {n, 1, 23}]
nmax = 23; CoefficientList[Series[Sum[x^k/(1 - 6 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 6^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339686(n): return sum(6^(k-1) for k in (1..n) if (k).divides(n))
[A339686(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 6*x^k).
G.f.: Sum_{k>=1} 6^(k-1) * x^k / (1 - x^k).
a(n) ~ 6^(n-1). - Vaclav Kotesovec, Jun 05 2021

A339687 a(n) = Sum_{d|n} 7^(d-1).

Original entry on oeis.org

1, 8, 50, 351, 2402, 16864, 117650, 823894, 5764851, 40356016, 282475250, 1977343950, 13841287202, 96889128064, 678223075300, 4747562333837, 33232930569602, 232630519768872, 1628413597910450, 11398895225729502, 79792266297729700, 558545864365759264

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 7 of A308813.
Cf. A000420, A320072.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), this sequence (q=7), A339688 (q=8), A339689 (q=9).

A339687:= func< n | (&+[7^(d-1): d in Divisors(n)]) >;
[A339687(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[7^(d - 1), {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/(1 - 7 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 7^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339687(n): return sum(7^(k-1) for k in (1..n) if (k).divides(n))
[A339687(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 7*x^k).
G.f.: Sum_{k>=1} 7^(k-1) * x^k / (1 - x^k).
a(n) ~ 7^(n-1). - Vaclav Kotesovec, Jun 05 2021

A339688 a(n) = Sum_{d|n} 8^(d-1).

Original entry on oeis.org

1, 9, 65, 521, 4097, 32841, 262145, 2097673, 16777281, 134221833, 1073741825, 8589967945, 68719476737, 549756076041, 4398046515265, 35184374186505, 281474976710657, 2251799830495305, 18014398509481985, 144115188210078217, 1152921504607109185

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 8 of A308813.
Cf. A001018, A320073.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), this sequence (q=8), A339689 (q=9).

A339688:= func< n | (&+[8^(d-1): d in Divisors(n)]) >;
[A339688(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[8^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 8 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 8^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339688(n): return sum(8^(k-1) for k in (1..n) if (k).divides(n))
[A339688(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 8*x^k).
G.f.: Sum_{k>=1} 8^(k-1) * x^k / (1 - x^k).
a(n) ~ 8^(n-1). - Vaclav Kotesovec, Jun 05 2021

A339689 a(n) = Sum_{d|n} 9^(d-1).

Original entry on oeis.org

1, 10, 82, 739, 6562, 59140, 531442, 4783708, 43046803, 387427060, 3486784402, 31381119478, 282429536482, 2541866359780, 22876792461604, 205891136878357, 1853020188851842, 16677181742772430, 150094635296999122, 1350851718060419878, 12157665459057460324

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 9 of A308813.
Cf. A001019, A320074.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), this sequence (q=9).

A339689:= func< n | (&+[9^(d-1): d in Divisors(n)]) >;
[A339689(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[9^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 9 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 9^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339689(n): return sum(9^(k-1) for k in (1..n) if (k).divides(n))
[A339689(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 9*x^k).
G.f.: Sum_{k>=1} 9^(k-1) * x^k / (1 - x^k).
a(n) ~ 9^(n-1). - Vaclav Kotesovec, Jun 05 2021

A308813 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) is Sum_{d|n} k^(d-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 5, 3, 1, 1, 5, 10, 11, 2, 1, 1, 6, 17, 31, 17, 4, 1, 1, 7, 26, 69, 82, 39, 2, 1, 1, 8, 37, 131, 257, 256, 65, 4, 1, 1, 9, 50, 223, 626, 1045, 730, 139, 3, 1, 1, 10, 65, 351, 1297, 3156, 4097, 2218, 261, 4, 1

Offset: 1

Seiichi Manyama, Jun 26 2019

			Square array, A(n,k), begins:
  1, 1,  1,   1,    1,     1,     1, ...
  1, 2,  3,   4,    5,     6,     7, ...
  1, 2,  5,  10,   17,    26,    37, ...
  1, 3, 11,  31,   69,   131,   223, ...
  1, 2, 17,  82,  257,   626,  1297, ...
  1, 4, 39, 256, 1045,  3156,  7819, ...
  1, 2, 65, 730, 4097, 15626, 46657, ...
Antidiagonal triangle, T(n,k), begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  2,   1;
  1,  4,  5,   3,    1;
  1,  5, 10,  11,    2,    1;
  1,  6, 17,  31,   17,    4,    1;
  1,  7, 26,  69,   82,   39,    2,    1;
  1,  8, 37, 131,  257,  256,   65,    4,   1;
  1,  9, 50, 223,  626, 1045,  730,  139,   3,   1;
  1, 10, 65, 351, 1297, 3156, 4097, 2218, 261,   4,   1;

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..10 give A000012, A000005, A034729, A034730, A339684, A339685, A339686, A339687, A339688, A339689, A113999.
Row n=1..3 give A000012, A000027(k+1), A002522.
A(n,n) gives A308814.

A:= func< n,k | (&+[k^(d-1): d in Divisors(n)]) >;
A308813:= func< n,k | A(k+1,n-k-1) >;
[A308813(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 26 2024

A[n_, k_] := DivisorSum[n, If[k == # - 1 == 0, 1, k^(# - 1)] &];
Table[A[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, May 07 2021 *)

def A(n,k): return sum(k^(j-1) for j in (1..n) if (j).divides(n))
def A308813(n,k): return A(k+1,n-k-1)
flatten([[A308813(n,k) for k in range(n)] for n in range(1,13)]) # G. C. Greubel, Jun 26 2024

G.f. of column k: Sum_{j>=1} x^j/(1 - k*x^j).

T(n, k) = Sum_{d|(k+1)} (n-k-1)^(d-1), with T(n, n) = 1. - G. C. Greubel, Jun 26 2024

A113998 Reverse of triangle A051731.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1

Offset: 1

Paul Barry, Nov 12 2005

Row sums are A000005. Diagonal sums are A001227 (?). Row k corresponds to A113999(k).

			Triangle begins
1;
1,1;
1,0,1;
1,0,1,1;
1,0,0,0,1;
1,0,0,1,1,1;
1,0,0,0,0,0,1;
1,0,0,0,1,0,1,1;

Sum_{k, 1<=k<=n} k*T(n,k)=A081307(n) - Philippe Deléham, Feb 03 2007

cp's OEIS Frontend

A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

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A034730 Dirichlet convolution of b_n=1 with c_n=3^(n-1).

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A339684 a(n) = Sum_{d|n} 4^(d-1).

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A339685 a(n) = Sum_{d|n} 5^(d-1).

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A339686 a(n) = Sum_{d|n} 6^(d-1).

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A339687 a(n) = Sum_{d|n} 7^(d-1).

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Magma

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A339688 a(n) = Sum_{d|n} 8^(d-1).