A000312 -id:A000312 - cp's OEIS frontend

A062796 Inverse Moebius transform of f(n) = n^n (A000312).

1, 5, 28, 261, 3126, 46688, 823544, 16777477, 387420517, 10000003130, 285311670612, 8916100495200, 302875106592254, 11112006826381564, 437893890380862528, 18446744073726329093, 827240261886336764178, 39346408075296925042601, 1978419655660313589123980

Offset: 1

Labos Elemer, Jul 19 2001

nonn

			n=6: divisors = {1,2,3,6}; 1^1 + 2^2 + 3^3 + 6^6 = 1 + 4 + 27 + 46656 = 46688 = a(6).

Seiichi Manyama, Table of n, a(n) for n = 1..386 (first 200 terms from Nick Hobson)

Cf. A000312, A023879.

a[n_] := DivisorSum[n, #^# &]; Array[a, 19] (* Jean-François Alcover, Dec 23 2015 *)

PARI
```
vector(17, n, sumdiv(n, d, d^d))
```

{a(n)=polcoeff(sum(m=1,n,m^m*x^m/(1-x^m +x*O(x^n))),n)} \\ Paul D. Hanna, Oct 27 2009

a(n) = sumdiv(n,d, d^d ); \\ Joerg Arndt, Apr 14 2013

from sympy import divisors
def A062796(n): return sum(d**d for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

a(n) = Sum_{d|n} d^d.
G.f.: Sum_{n>=1} n^n * x^n/(1 - x^n). - Paul D. Hanna, Oct 27 2009
Logarithmic derivative of A023879. - Paul D. Hanna, Sep 05 2012

A062319 Number of divisors of n^n, or of A000312(n).

Original entry on oeis.org

1, 1, 3, 4, 9, 6, 49, 8, 25, 19, 121, 12, 325, 14, 225, 256, 65, 18, 703, 20, 861, 484, 529, 24, 1825, 51, 729, 82, 1653, 30, 29791, 32, 161, 1156, 1225, 1296, 5329, 38, 1521, 1600, 4961, 42, 79507, 44, 4005, 4186, 2209, 48, 9457, 99, 5151, 2704, 5565, 54

Offset: 0

Jason Earls, Jul 05 2001

From Gus Wiseman, May 02 2021: (Start)
Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30. For example, the a(1) = 1 through a(5) = 6 tuples are:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
       (1,2)  (1,1,3)  (1,1,1,2)  (1,1,1,1,5)
       (2,1)  (1,3,1)  (1,1,1,4)  (1,1,1,5,1)
              (3,1,1)  (1,1,2,1)  (1,1,5,1,1)
                       (1,1,4,1)  (1,5,1,1,1)
                       (1,2,1,1)  (5,1,1,1,1)
                       (1,4,1,1)
                       (2,1,1,1)
                       (4,1,1,1)
The unordered case (pairwise coprime n-multisets of divisors of n) is counted by A343654.
(End)

			From _Gus Wiseman_, May 02 2021: (Start)
The a(1) = 1 through a(5) = 6 divisors:
  1  1  1   1    1
     2  3   2    5
     4  9   4    25
        27  8    125
            16   625
            32   3125
            64
            128
            256
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Cf. A046798, A077592, A035116. - Enrique Pérez Herrero, Nov 09 2010
Number of divisors of A000312(n).
Taking Omega instead of sigma gives A066959.
Positions of squares are A173339.
Diagonal n = k of the array A343656.
A000005 counts divisors.
A059481 counts k-multisets of elements of {1..n}.
A334997 counts length-k strict chains of divisors of n.
A343658 counts k-multisets of divisors.
Pairwise coprimality:
- A018892 counts coprime pairs of divisors.
- A084422 counts pairwise coprime subsets of {1..n}.
- A100565 counts pairwise coprime triples of divisors.
- A225520 counts pairwise coprime sets of divisors.
- A343652 counts maximal pairwise coprime sets of divisors.
- A343653 counts pairwise coprime non-singleton sets of divisors > 1.
- A343654 counts pairwise coprime sets of divisors > 1.
Cf. A000169, A000272, A002064, A002109, A009998, A048691, A143773, A146291, A176029, A327527, A343657.

[NumberOfDivisors(n^n): n in  [0..60]]; // Vincenzo Librandi, Nov 09 2014

A062319[n_IntegerQ]:=DivisorSigma[0,n^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
Join[{1},DivisorSigma[0,#^#]&/@Range[60]] (* Harvey P. Dale, Jun 06 2024 *)

je=[]; for(n=0,200,je=concat(je,numdiv(n^n))); je

{ for (n=0, 1000, write("b062319.txt", n, " ", numdiv(n^n)); ) } \\ Harry J. Smith, Aug 04 2009

a(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],fm[k,2]*n+1) \\ Franklin T. Adams-Watters, May 03 2011

a(n) = if(n==0, 1, sumdiv(n, d, n^omega(d))); \\ Seiichi Manyama, May 12 2021

from math import prod
from sympy import factorint
def A062319(n): return prod(n*d+1 for d in factorint(n).values()) # Chai Wah Wu, Jun 03 2021

a(n) = A000005(A000312(n)). - Enrique Pérez Herrero, Nov 09 2010
a(2^n) = A002064(n). - Gus Wiseman, May 02 2021
a(prime(n)) = prime(n) + 1. - Gus Wiseman, May 02 2021
a(n) = Product_{i=1..s} (1 + n * m_i) where (m_1,...,m_s) is the sequence of prime multiplicities (prime signature) of n. - Gus Wiseman, May 02 2021
a(n) = Sum_{d|n} n^omega(d) for n > 0. - Seiichi Manyama May 12 2021

A062727 Sum of the divisors of n^n (A000312).

Original entry on oeis.org

1, 1, 7, 40, 511, 3906, 138811, 960800, 33554431, 581130733, 24987792457, 313842837672, 26748283770391, 328114698808274, 25927224666044919, 821051025385244160, 36893488147419103231, 878942778254232811938

Offset: 0

Jason Earls, Jul 11 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000312.

Table[DivisorSigma[1,n^n],{n,1,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

PARI
```
for(n=0,22,print(sigma(n^n)))
```

{ for (n=0, 100, write("b062727.txt", n, " ", sigma(n^n)) ) } \\ Harry J. Smith, Aug 09 2009

a(n) = A000203(A000312(n)). - Michel Marcus, Jan 10 2015

A100262 Expansion of A(x)^2, where A(x) = o.g.f. of n^n (A000312).

Original entry on oeis.org

1, 2, 9, 62, 582, 6978, 102339, 1779222, 35809052, 819103178, 20987183525, 595341928814, 18519658804818, 626784970780690, 22926284614808071, 901188628763393606, 37882728189752349304, 1695744102631158083866

Offset: 0

Ralf Stephan, Nov 20 2004

nonn

			(1 + x + 4x^2 + 27x^3 + 256x^4 +...)^2 = 1 + 2x + 9x^2 + 62x^3 +...

Seiichi Manyama, Table of n, a(n) for n = 0..386
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

Cf. A000312, A053729, A062817, A349874.

nn=17;f[x_]=1+Sum[n^n x^n,{n,1,nn}];CoefficientList[Series[f[x]^2,{x,0,nn}],x] (* Geoffrey Critzer, Nov 05 2013 *)

a(n) = sum(k=0, n, k^k*(n-k)^(n-k)); \\ Seiichi Manyama, Dec 03 2021

a(n) = Sum_{k=0..n} k^k * (n-k)^(n-k). - Tilman Neumann, Dec 13 2008

a(n) ~ 2 * n^n. - Vaclav Kotesovec, Dec 03 2021

A156223 Numbers that are products of distinct terms in A000312.

Original entry on oeis.org

1, 4, 27, 108, 256, 1024, 3125, 6912, 12500, 27648, 46656, 84375, 186624, 337500, 800000, 823543, 1259712, 3200000, 3294172, 5038848, 11943936, 16777216, 21600000, 22235661, 47775744, 67108864, 86400000, 88942644, 145800000, 210827008, 322486272, 387420489, 452984832

Offset: 1

Gabriel C. Benamy, Feb 06 2009

nonn

			a(1) = 1 = 1^1.
a(2) = 4 = 2^2.
a(3) = 108 = 2^2 * 3^3.
...
a(20) = 5038848 = 2^2 * 3^3 * 6^6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000312.

seq[max_] := Module[{kmax = 1, s = {1}}, While[kmax^kmax < max, kmax++]; Do[s = Select[Union[s, k^k*s], # <= max &], {k, 1, kmax}]; s]; seq[10^8] (* Amiram Eldar, Nov 24 2020 *)

More terms from Amiram Eldar, Nov 24 2020

A326985 G.f.: B(x)B(2x^2)B(3x^3)*..., where B(x) is g.f. of A000312.

Original entry on oeis.org

1, 1, 6, 32, 287, 3222, 47606, 831488, 16890792, 389286222, 10037183606, 286154919078, 8937624574652, 303483905672078, 11130904101218094, 438532313635906858, 18470060947222927499, 828155619735377936654, 39384843256547964375436, 1980138439071577626157382

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000312, A110143, A326986.

B:= proc(n) option remember; n^n end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      B(n), add(b(j, 1)*i^j*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 23 2019

nmax = 20; CoefficientList[Series[Product[1+Sum[k^k*j^k*x^(j*k), {k, 1, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ n^n.

A053729 Self-convolution of 1,4,27,256,3125,46656,... (cf. A000312).

Original entry on oeis.org

1, 8, 70, 728, 9027, 132136, 2254620, 44262200, 987183525, 24718587592, 687457908306, 21034757596184, 702270963692039, 25400848001674856, 989240042333246072, 41263578858484555512, 1835070614332428285513

Offset: 1

Leroy Quet, Feb 11 2000

			a(4) = 1^1 *4^4 +2^2 *3^3 +3^3 *2^2 +4^4 *1^1 = 1*256 +4*27 +27*4 +256*1 = 728.

John Tyler Rascoe, Table of n, a(n) for n = 1..100

Cf. A000312, A100262.

nn=20;f[x_]=Sum[n^n x^n,{n,1,nn}];CoefficientList[Series[f[x]^2/x^2,{x,0,nn}],x] (* Geoffrey Critzer, Nov 05 2013 *)
Table[Sum[k^k*(n+1-k)^(n+1-k), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 10 2018 *)

def A053729(n): return sum((k**k)*(n+1-k)**(n+1-k) for k in range(1,n+1)) # John Tyler Rascoe, Aug 23 2024

a(n) = Sum_{k=1..n} k^k * (n+1-k)^(n+1-k).

a(n) ~ 2 * n^n. - Vaclav Kotesovec, Mar 10 2018

More terms from James Sellers, Feb 22 2000

A089901 Main diagonal of A089900, also the inverse hyperbinomial of A000312 (offset 1).

Original entry on oeis.org

1, 3, 18, 159, 1848, 26595, 456048, 9073911, 205437312, 5214027267, 146602156800, 4522866752943, 151895344131072, 5516066815430691, 215373243256915968, 8996883483108522375, 400372897193449586688, 18908951043963993686019

Offset: 0

Paul D. Hanna, Nov 14 2003

nonn

The n-th row of array A089900 is the n-th binomial transform of the factorials found in row 0: {1!,2!,3!,...,(n+1)!,...}. The hyperbinomial transform of this main diagonal gives: {1,4,27,...,(n+1)^(n+1),...}, which is the next lower diagonal in array A089900.

a(n), for n>=1, is the number of colored labeled mappings from n points to themselves, where each component is one of three colors. - Steven Finch, Nov 28 2021

G. C. Greubel, Table of n, a(n) for n = 0..380
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 162.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Cf. A000312, A089900, A089902, A000312.

CoefficientList[Series[1/(1+LambertW[-x])^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
Flatten[{1,Table[Sum[n^(n-k)*Binomial[n,k]*(k+1)!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 09 2013 *)
a[n_] := (n^(n + 2) + Exp[n] Gamma[n + 2, n]) / (n + 1);
Table[a[n], {n, 0, 17}]  (* Peter Luschny, Nov 29 2021 *)

/* As (n+1)-th term of the n-th binomial transform of {(n+1)!}: */
{a(n)=if(n<0,0,sum(i=0,n,n^(n-i)*binomial(n,i)*(i+1)!))}
for(n=0,20,print1(a(n),", "))

/* As (n+1)-th term of inverse hyperbinomial of {(n+1)^(n+1)}: */
{a(n)=if(n<0,0,sum(i=0,n,-(n-i-1)^(n-i-1)*binomial(n,i)*(i+1)^(i+1)))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} n^(n-k) * C(n,k) * (k+1)!.
a(n) = Sum_{k=0..n} -(n-k-1)^(n-k-1) * C(n,k) * (k+1)^(k+1).
E.g.f.: 1 / (1 + LambertW(-x))^3.
E.g.f.: (Sum_{n>=0} (n+1)^(n+1) * x^n/n!) * (Sum_{n>=0} -(n-1)^(n-1) * x^n/n!).
a(n) ~ n^(n+1) * (1 + sqrt(Pi/(2*n))). - Vaclav Kotesovec, Jul 09 2013
a(n) = (n^(n + 2) + exp(n)*Gamma(n + 2, n)) / (n + 1). - Peter Luschny, Nov 29 2021

A309652 a(n) = [x^n] B(x)^n, where B(x) is g.f. of A000312.

Original entry on oeis.org

1, 1, 9, 106, 1493, 24276, 448122, 9301251, 215547845, 5541171496, 156997349684, 4870353700532, 164366482285898, 5998207807965543, 235388194276592723, 9884482616014596546, 442206843338189113445, 20995082225203329126384, 1054247070579064423466016

Offset: 0

Vaclav Kotesovec, Aug 11 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..385

Cf. A287899, A326985, A326986.

B:= proc(n) option remember; n^n end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, B(n),
      (h-> add(b(j, h)*b(n-j, i-h), j=0..n))(iquo(i, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 23 2019

Table[SeriesCoefficient[(1+Sum[k^k*x^k, {k, 1, n}])^n, {x, 0, n}], {n, 0, 20}]

a(n) ~ exp(exp(-1)) * n^(n+1).

A326986 G.f.: B(x)B(x^2)B(x^3)*..., where B(x) is g.f. of A000312.

Original entry on oeis.org

1, 1, 5, 29, 266, 3163, 46994, 827107, 16828741, 388308078, 10017853262, 285720195351, 8926575094978, 303172417424680, 11121259586618456, 438207141286916539, 18458204444260001120, 827690809585441201775, 39365349178064541861252, 1979267564496263599093676

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000312, A096161, A309652, A326985.

B:= proc(n) option remember; n^n end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      B(n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 23 2019

nmax = 20; CoefficientList[Series[Product[1+Sum[k^k*x^(j*k), {k, 1, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ n^n.

cp's OEIS Frontend

A062796 Inverse Moebius transform of f(n) = n^n (A000312).

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A062319 Number of divisors of n^n, or of A000312(n).

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A062727 Sum of the divisors of n^n (A000312).

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A100262 Expansion of A(x)^2, where A(x) = o.g.f. of n^n (A000312).

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Mathematica

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A156223 Numbers that are products of distinct terms in A000312.

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A326985 G.f.: B(x)*B(2*x^2)*B(3*x^3)*..., where B(x) is g.f. of A000312.

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A053729 Self-convolution of 1,4,27,256,3125,46656,... (cf. A000312).

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A326985 G.f.: B(x)B(2x^2)B(3x^3)*..., where B(x) is g.f. of A000312.

A326986 G.f.: B(x)B(x^2)B(x^3)*..., where B(x) is g.f. of A000312.