A034729 -id:A034729 - cp's OEIS frontend

A047968 a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d.

1, 3, 4, 8, 8, 17, 16, 30, 34, 52, 57, 99, 102, 153, 187, 261, 298, 432, 491, 684, 811, 1061, 1256, 1696, 1966, 2540, 3044, 3876, 4566, 5846, 6843, 8610, 10203, 12610, 14906, 18491, 21638, 26508, 31290, 38044, 44584, 54133, 63262, 76241

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

Inverse Moebius transform of A000041.
Row sums of triangle A137587. - Gary W. Adamson, Jan 27 2008
Row sums of triangle A168021. - Omar E. Pol, Nov 20 2009
Row sums of triangle A168017. Row sums of triangle A168018. - Omar E. Pol, Nov 25 2009
Sum of the partition numbers of the divisors of n. - Omar E. Pol, Feb 25 2014
Conjecture: for n > 6, a(n) is strictly increasing. - Franklin T. Adams-Watters, Apr 19 2014
Number of constant multiset partitions of multisets spanning an initial interval of positive integers with multiplicities an integer partition of n. - Gus Wiseman, Sep 16 2018

			For n = 10 the divisors of 10 are 1, 2, 5, 10, hence the partition numbers of the divisors of 10 are 1, 2, 7, 42, so a(10) = 1 + 2 + 7 + 42 = 52. - _Omar E. Pol_, Feb 26 2014
From _Gus Wiseman_, Sep 16 2018: (Start)
The a(6) = 17 constant multiset partitions:
  (111111)  (111)(111)    (11)(11)(11)  (1)(1)(1)(1)(1)(1)
  (111222)  (12)(12)(12)
  (111122)  (112)(112)
  (112233)  (123)(123)
  (111112)
  (111123)
  (111223)
  (111234)
  (112234)
  (112345)
  (123456)
(End)

T. D. Noe, Table of n, a(n) for n=1..1000
N. J. A. Sloane, Transforms

Cf. A000041, A000837, A047966, A055893, A137587, A003606 (Euler transform).

Cf. A002033, A003238, A018783, A034729, A052409, A078392, A100953, A319162.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l) do c := c+numbpart(l[i]) od: RETURN(c): end: for j from 1 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007

a[n_] := Sum[ PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Oct 03 2013 *)

G.f.: Sum_{k>0} (-1+1/Product_{i>0} (1-z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: sum(n>0,A000041(n)*x^n/(1-x^n)). - Mircea Merca, Feb 24 2014.
a(n) = A168111(n) + A000041(n). - Omar E. Pol, Feb 26 2014
a(n) = Sum_{y is a partition of n} A000005(GCD(y)). - Gus Wiseman, Sep 16 2018

A320456 Numbers whose multiset multisystem spans an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 61, 63, 64, 65, 69, 70, 72, 74, 75, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The n-th multiset multisystem is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the 78th multiset multisystem is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}

Cf. A001222, A003963, A034691, A034729, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320458, A320459, A320461, A320462, A320463, A320464, A320532, A320533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[100],normQ[primeMS/@primeMS[#]]&]

A074854 a(n) = Sum_{d|n} (2^(n-d)).

Original entry on oeis.org

1, 3, 5, 13, 17, 57, 65, 209, 321, 801, 1025, 3905, 4097, 12417, 21505, 53505, 65537, 233985, 262145, 885761, 1327105, 3147777, 4194305, 16060417, 17825793, 50339841, 84148225, 220217345, 268435457, 990937089, 1073741825, 3506503681

Offset: 1

Miklos Kristof, Sep 11 2002

A034729 = Sum_{d|n} (2^(d-1)).
A055895 = 2*A034729.
If p is a prime, then a(p) = A034729(p) = 2^(p-1)+1.
From Gus Wiseman, Jul 14 2020: (Start)
Number of ways to tile a rectangle of size n using horizontal strips. Also the number of ways to choose a composition of each part of a constant partition of n. The a(0) = 1 through a(5) = 17 splittings are:
  ()  (1)  (2)      (3)          (4)              (5)
           (1,1)    (1,2)        (1,3)            (1,4)
           (1),(1)  (2,1)        (2,2)            (2,3)
                    (1,1,1)      (3,1)            (3,2)
                    (1),(1),(1)  (1,1,2)          (4,1)
                                 (1,2,1)          (1,1,3)
                                 (2,1,1)          (1,2,2)
                                 (2),(2)          (1,3,1)
                                 (1,1,1,1)        (2,1,2)
                                 (1,1),(2)        (2,2,1)
                                 (2),(1,1)        (3,1,1)
                                 (1,1),(1,1)      (1,1,1,2)
                                 (1),(1),(1),(1)  (1,1,2,1)
                                                  (1,2,1,1)
                                                  (2,1,1,1)
                                                  (1,1,1,1,1)
                                                  (1),(1),(1),(1),(1)
(End)

			Divisors of 6 = 1,2,3,6 and 6-1 = 5, 6-2 = 4, 6-3 = 3, 6-6 = 0. a(6) = 2^5 + 2^4 + 2^3 + 2^0 = 32 + 16 + 8 + 1 = 57.
G.f. = x + 3*x^2 + 5*x^3 + 13*x^4 + 17*x^5 + 57*x^6 + 65*x^7 + ...
a(14) = 1 + 2^7 + 2^12 + 2^13 = 12417. - _Gus Wiseman_, Jun 20 2018

Gus Wiseman, Table of n, a(n) for n = 1..32

Cf. A055895, A034729.
Cf. A080267.
Cf. A051731.
The version looking at lengths instead of sums is A101509.
The strictly increasing (or strictly decreasing) version is A304961.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Requiring distinct instead of equal sums gives A336127.
Starting with a strict composition gives A336130.
Partitions of partitions are A001970.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Cf. A006951, A063834, A075900, A279787, A305551, A316245, A323433, A336128.

a[ n_] := If[ n < 1, 0, Sum[ 2^(n - d), {d, Divisors[n]}]] (* Michael Somos, Mar 28 2013 *)

a(n)=if(n<1,0,2^n*polcoeff(sum(k=1,n,2/(2-x^k),x*O(x^n)),n))

a(n) = sumdiv(n,d, 2^(n-d) ); /* Joerg Arndt, Mar 28 2013 */

G.f.: 2^n times coefficient of x^n in Sum_{k>=1} x^k/(2-x^k). - Benoit Cloitre, Apr 21 2003; corrected by Joerg Arndt, Mar 28 2013
G.f.: Sum_{k>0} 2^(k-1)*x^k/(1-2^(k-1)*x^k). - Vladeta Jovovic, Jun 24 2003
G.f.: Sum_{n>=1} a*z^n/(1-a*z^n) (generalized Lambert series) where z=2*x and a=1/2. - Joerg Arndt, Jan 30 2011
Triangle A051731 mod 2 converted to decimal. - Philippe Deléham, Oct 04 2003
G.f.: Sum_{k>0} 1 / (2 / (2*x)^k - 1). - Michael Somos, Mar 28 2013

a(14) corrected from 9407 to 12417 by Gus Wiseman, Jun 20 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

A268235 a(n) = Sum_{k=1..n} floor(n/k)*2^(k-1).

Original entry on oeis.org

1, 4, 9, 20, 37, 76, 141, 280, 541, 1072, 2097, 4192, 8289, 16548, 32953, 65860, 131397, 262764, 524909, 1049736, 2098381, 4196560, 8390865, 16781696, 33558929, 67117460, 134226585, 268452580, 536888037, 1073775900, 2147517725, 4295034280, 8590002605, 17180002736, 34359872001, 68719743792

Offset: 1

Benoit Cloitre and N. J. A. Sloane, Feb 05 2016

nonn

This is the "floor transform" of the powers of 2.

Matthew House, Table of n, a(n) for n = 1..3305

First differences give A034729.
Cf. A000079.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), this sequence (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A268235:= func< n | (&+[Floor(n/j)*2^(j-1): j in [1..n]]) >;
[A268235(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024

# floor transform of a sequence
ft:=proc(a) local b,n,j,k; b:=[];
for n from 1 to nops(a) do j:=add(a[k]*floor(n/k),k=1..n); b:=[op(b),j]; od;
b; end:
ft([seq(2^i,i=0..50)]);

Table[Sum[Floor[n/k] 2^(k - 1), {k, n}], {n, 36}] (* Michael De Vlieger, Feb 12 2017 *)

a(n) = sum(k=1, n, (n\k)*2^(k-1)); \\ Michel Marcus, Feb 11 2017

a(n) = sum(k=1, n, sumdiv(k, d, 2^(d-1))); \\ Seiichi Manyama, May 29 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-2*x^k))/(1-x)) \\ Seiichi Manyama, May 29 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 2^(k-1)*x^k/(1-x^k))/(1-x)) \\ Seiichi Manyama, May 29 2021

def A268235(n): return sum((n//j)*2^(j-1) for j in range(1,n+1))
[A268235(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

a(n) ~ 2^n. - Vaclav Kotesovec, May 28 2021
From Seiichi Manyama, May 29 2021: (Start)
a(n) = Sum_{k=1..n} Sum_{d|k} 2^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 2*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 2^(k-1) * x^k/(1 - x^k). (End)
a(n) = Sum_{k=1..n} (2^floor(n/k) - 1). - Ridouane Oudra, Feb 03 2023

Definition corrected by Matthew House, Feb 11 2017

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

A055895 Inverse Moebius transform of powers of 2.

Original entry on oeis.org

1, 2, 6, 10, 22, 34, 78, 130, 278, 522, 1062, 2050, 4190, 8194, 16518, 32810, 65814, 131074, 262734, 524290, 1049654, 2097290, 4196358, 8388610, 16781662, 33554466, 67117062, 134218250, 268451990, 536870914, 1073775726, 2147483650, 4295033110, 8589936650

Offset: 0

Christian G. Bower, Jun 09 2000

nonn

Row sums of A055894.

			G.f. = 1 + 2*x + 6*x^2 + 10*x^3 + 22*x^4 + 34*x^5 + 78*x^6 + 130*x^7 + ...

T. D. Noe, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A034729, A113705 (binary), A339916.

Cf. A055894.

Table[Plus @@ Map[Function[d, 2^d], Divisors[n]], {n, 0, 30}] (* Olivier Gérard, Jan 01 2012 *)
a[0]=1; a[n_] := DivisorSum[n, 2^#&]; Array[a, 40, 0] (* Jean-François Alcover, Dec 01 2015 *)

a(n)=if(n<1,1,polcoeff(sum(k=1,n,1/(1-2*x^k),x*O(x^n)),n))

a(n)=if(n<1,1,sumdiv(n,d,2^d)); /* Joerg Arndt, Aug 14 2012 */

G.f.: 1 + Sum_{k>=1} 2^k*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
a(n) = Sum_{d divides n} 2^d. - Olivier Gérard, Jan 01 2012
a(n) = 2 * A034729(n) for n >= 1. - Joerg Arndt, Aug 14 2012
G.f.: 1 + Sum_{k>=1} 2*x^k/(1-2*x^k). - Joerg Arndt, Mar 28 2013

A113999 a(n) = Sum_{ k, k|n } 10^(k-1).

Original entry on oeis.org

1, 11, 101, 1011, 10001, 100111, 1000001, 10001011, 100000101, 1000010011, 10000000001, 100000101111, 1000000000001, 10000001000011, 100000000010101, 1000000010001011, 10000000000000001, 100000000100100111

Offset: 1

Paul Barry, Nov 12 2005

A034729 to base 2. Stacking elements of the sequence gives A113998.

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

Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), this sequence (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A113999:= func< n | (&+[10^(d-1): d in Divisors(n)]) >;
[A113999(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

A113999[n_]:= DivisorSum[n, 10^(#-1) &];
Table[A113999[n], {n, 40}] (* G. C. Greubel, Jun 26 2024 *)

PARI
```
a(n)=if(n<1,0,sumdiv(n,k,10^(k-1)));
```

def A113999(n): return sum(10^(k-1) for k in (1..n) if (k).divides(n))
[A113999(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

G.f.: Sum_{n>0} x^n/(1-10*x^n).

a(n) ~ 10^(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

A382204 Number of normal multiset partitions of weight n into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 5, 8, 8, 10, 8, 15, 9, 14, 15, 17, 13, 22, 14, 25, 21, 23, 19, 34, 24, 29, 28, 37, 27, 45, 29, 44, 38, 43, 43, 59, 40, 51, 48, 69, 48, 71, 52, 73, 69, 72, 61, 93, 72, 91, 77, 99, 78, 105, 95, 119, 95, 113, 96, 146, 107, 126, 123, 151, 130

Offset: 0

Gus Wiseman, Mar 26 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(6) = 7 multiset partitions:
  {1} {11}   {111}     {1111}       {11111}         {111111}
      {1}{1} {2}{11}   {11}{11}     {2}{11}{11}     {111}{111}
             {1}{1}{1} {2}{2}{11}   {2}{2}{2}{11}   {22}{1111}
                       {1}{1}{1}{1} {1}{1}{1}{1}{1} {11}{11}{11}
                                                    {2}{2}{11}{11}
                                                    {2}{2}{2}{2}{11}
                                                    {1}{1}{1}{1}{1}{1}
The a(1) = 1 through a(7) = 5 factorizations:
  2  4    8      16       32         64           128
     2*2  3*4    4*4      3*4*4      8*8          3*4*4*4
          2*2*2  3*3*4    3*3*3*4    9*16         3*3*3*4*4
                 2*2*2*2  2*2*2*2*2  4*4*4        3*3*3*3*3*4
                                     3*3*4*4      2*2*2*2*2*2*2
                                     3*3*3*3*4
                                     2*2*2*2*2*2

Christian Sievers, Table of n, a(n) for n = 0..25000

Without a common sum we have A055887.
Twice-partitions of this type are counted by A279789.
Without constant blocks we have A326518.
For distinct block-sums and strict blocks we have A381718.
Factorizations of this type are counted by A381995.
For distinct instead of equal block-sums we have A382203.
For strict instead of constant blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A255906 counts normal multiset partitions, row sums of A317532.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A304969, A356945.
Set multipartitions: A116540, A270995, A296119, A318360.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Constant blocks with distinct sums: A381635, A381636, A381716.
Cf. A007716, A034691, A034729, A116539, A255903, A317583, A326520, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],SameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

h(s,x)=my(t=0,p=1,k=1);while(s%k==0,p*=1/(1-x^(s/k))-1;t+=p;k+=1);t
lista(n)=Vec(1+sum(s=1,n,h(s,x+O(x*x^n)))) \\ Christian Sievers, Apr 05 2025

G.f.: 1 + Sum_{s>=1} Sum_{k=1..A055874(s)} Product_{v=1..k} (1/(1-x^(s/v)) - 1). - Christian Sievers, Apr 05 2025

Terms a(16) and beyond from Christian Sievers, Apr 04 2025

cp's OEIS Frontend

A047968 a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d.

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Maple

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A320456 Numbers whose multiset multisystem spans an initial interval of positive integers.

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Mathematica

A074854 a(n) = Sum_{d|n} (2^(n-d)).

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PARI

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

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A268235 a(n) = Sum_{k=1..n} floor(n/k)*2^(k-1).

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

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A055895 Inverse Moebius transform of powers of 2.

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