A023037 -id:A023037 - cp's OEIS frontend

A294316 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where A(n,k) is the sum of products of terms in all partitions of k*n into powers of k.

1, 1, 1, 1, 3, 1, 1, 4, 11, 1, 1, 5, 13, 27, 1, 1, 6, 21, 49, 83, 1, 1, 7, 31, 85, 157, 195, 1, 1, 8, 43, 156, 357, 481, 515, 1, 1, 9, 57, 259, 781, 1445, 1534, 1155, 1, 1, 10, 73, 400, 1555, 3931, 5797, 4693, 2899, 1, 1, 11, 91, 585, 2801, 9331, 19681, 23205

Offset: 0

Seiichi Manyama, Oct 28 2017

			Square array A(n,k) begins:
   1,   1,   1,    1,    1, ...
   1,   3,   4,    5,    6, ...
   1,  11,  13,   21,   31, ...
   1,  27,  49,   85,  156, ...
   1,  83, 157,  357,  781, ...
   1, 195, 481, 1445, 3931, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..3 give A000012, A289842, A294298.
Rows n=0 gives A000012.
Main diagonal gives A023037.
Cf. A292477.

A331672 Sum of all base-n numbers with digit sum n and length at most n.

Original entry on oeis.org

3, 91, 2635, 94501, 4254936, 234572213, 15403880115, 1176838159861, 102631111100848, 10063085278250005, 1095923297151849530, 131253123286275198027, 17145216226230367266330, 2425892898650501790637545, 369599184391990522425455939, 60326656013944234430010524773

Offset: 2

Alois P. Heinz, Feb 22 2020

The cardinality of these numbers is given by A048775(n-1).

			a(2) = 3 = 11_2.
a(3) = 91 = 5 + 7 + 11 + 13 + 15 + 19 + 21 = 12_3 + 21_3 + 102_3 + 111_3 + 120_3 + 201_3 + 210_3.
a(10) = A130835(10) = 102631111100848.

Alois P. Heinz, Table of n, a(n) for n = 2..322

Cf. A007953, A023037, A048775, A130835, A331673.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
      `if`(i=0, 0, add((p->[p[1], p[2]*k+p[1]*d])(
         b(n-d, i-1, k)), d=0..min(n, k-1))))
    end:
a:= n-> b(n$3)[2]:
seq(a(n), n=2..17);
# second Maple program:
a:= n-> (binomial(2*n-1, n)-n)*(n^n-1)/(n-1):
seq(a(n), n=2..17);

a(n) = A048775(n-1)*A023037(n) = (binomial(2*n-1,n)-n)*(n^n-1)/(n-1).

A332653 a(n) = (1/n) * Sum_{k=1..n} n^(k/gcd(n, k)).

Original entry on oeis.org

1, 2, 5, 19, 157, 1306, 19609, 266372, 5321721, 101001214, 2593742461, 61920391842, 1941507093541, 56984643437138, 2076518238897649, 72340172854919941, 3041324492229179281, 121440691499123469858, 5784852794328402307381, 262799364106291328009626

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Cf. A023037, A056665, A057661, A226561, A228640, A332620, A332621, A332652, A332655.

[(1/n)*&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020

Table[(1/n) Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^(k - 1), 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

a(n) = (1/n) * Sum_{k=1..n} n^(lcm(n, k)/n).
a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} n^(k-1).
a(n) = A332652(n) / n.

A368534 a(n) = Sum_{k=1..n} binomial(k+1,2) * n^(n-k).

Original entry on oeis.org

0, 1, 5, 24, 146, 1215, 13431, 186816, 3130436, 61291125, 1371742105, 34522712136, 964626945558, 29621465864627, 991330604373851, 35906022352657920, 1399219698628043016, 58367293868445147657, 2594796705962971336125, 122463905297217627859000

Offset: 0

Seiichi Manyama, Dec 29 2023

Cf. A062805, A368535.
Cf. A002662, A052150, A052161.
Cf. A023037, A062806, A068475.

Table[Sum[Binomial[k+1,2]n^(n-k),{k,n}],{n,0,20}] (* Harvey P. Dale, May 14 2025 *)

a(n) = sum(k=1, n, binomial(k+1, 2)*n^(n-k));

a(n) = [x^n] x/((1-n*x) * (1-x)^3).

a(n) = n * (2*n^(n+1) - n^3 - n^2 + n - 1)/(2 * (n-1)^3) for n > 1.

A068742 Largest power of n! which divides (n^n)!.

Original entry on oeis.org

1, 1, 8, 13060694016, 2079483819621516594237312893208982470178235671604741236585936787960152492743230443701290222268785125031295963705114624

Offset: 0

Henry Bottomley, Feb 26 2002

nonn

			a(3)=13060694016 since (3^3)!=27!=10888869450418352160768000000 can be divided by 3!=6 thirteen times and 6^13=13060694016.

Cf. A000142, A000312, A023037, A068740, A068741.

for(n=2,10,p=1:f=(n^n)!:while(f%p==0,p=p*n!):print1(p/n!","))

For p prime, a(p)=(p!)^((p^p-1)/(p-1))

More terms from Ralf Stephan, Mar 31 2003

A077385 Triangle read by rows in which n-th row contains 2n-1 terms starting from n^0 to n^(n-1) in increasing order and then in decreasing order to n^0.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 9, 3, 1, 1, 4, 16, 64, 16, 4, 1, 1, 5, 25, 125, 625, 125, 25, 5, 1, 1, 6, 36, 216, 1296, 7776, 1296, 216, 36, 6, 1, 1, 7, 49, 343, 2401, 16807, 117649, 16807, 2401, 343, 49, 7, 1, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 262144, 32768, 4096, 512, 64, 8, 1

Offset: 1

Amarnath Murthy, Nov 06 2002

			Irregular triangle begins as:
  1;
  1, 2,  1;
  1, 3,  9,   3,    1;
  1, 4, 16,  64,   16,     4,      1;
  1, 5, 25, 125,  625,   125,     25,     5,    1;
  1, 6, 36, 216, 1296,  7776,   1296,   216,   36,   6,  1;
  1, 7, 49, 343, 2401, 16807, 117649, 16807, 2401, 343, 49, 7, 1;

G. C. Greubel, Rows n = 1..40 of the irregular triangle, flattened

Cf. A000169, A000272, A023037, A060072, A077386.

A077385:= func< n,k | k lt n select n^k else n^(2*n-k-2) >;
[A077385(n,k): k in [0..2*n-2], n in [1..12]]; // G. C. Greubel, Sep 21 2022

A077385 := proc(n,k) if k < n then n^k ; else n^(2*n-k-2) ; fi ; end: for n from 1 to 10 do for k from 0 to 2*n-2 do printf("%d, ",A077385(n,k)) ; od : od : # R. J. Mathar, Jul 03 2007

Table[Join[n^Range[0,n-1],n^Range[n-2,0,-1]],{n,8}]//Flatten (* Harvey P. Dale, Oct 13 2017 *)

def A077385(n,k): return n^k if (kA077385(n,k) for k in (0..2*n-2)] for n in (1..12)]) # G. C. Greubel, Sep 21 2022

T(n, k) = n^k for k < n, otherwise n^(2*n-k-2), for n >= 1, 0 <= k <= 2*n-2.
From G. C. Greubel, Sep 21 2022: (Start)
T(n, 0) = T(n, 2*n-2) = 1.
T(n, n-1) = A000169(n).
T(n, n) = A000272(n).
T(n, 2*n-2-k) = T(n, k).
Sum_{k=0..n-1} T(n, k) = A023037(n).
Sum_{k=0..n-2} T(n, k) = A060072(n).
Sum_{k=0..2*n-2} T(n, k) = A077386(n) = 2*A060072(n) + A000169(n), n > 1. (End)

More terms from R. J. Mathar, Jul 03 2007

A117667 a(n) = n^n-n^(n-1)-n^(n-2)-n^(n-3)-...-n^3-n^2-n.

Original entry on oeis.org

1, 2, 15, 172, 2345, 37326, 686287, 14380472, 338992929, 8888888890, 256780503551, 8105545862052, 277635514376233, 10257237069745862, 406615755353655135, 17216961135462248176, 775537745518440716417

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 11 2006

nonn

			a(3) = 3^3-3^2-3 = 27-9-3 = 15.

Christian N. K. Anderson, Table of n, a(n) for n = 1..100

Cf. A000312 (n^n), A023037 (1+n+n^2+...n^(n-1)),

A001923, A031972.

a:=n->n^n-sum(n^j,j=1..n-1): seq(a(n),n=1..19); # Emeric Deutsch, Apr 16 2006

s[n_] := Sum[n^i, {i, 1, n - 1}]; Table[n^n - s[n], {n, 17}] (* Carlos Eduardo Olivieri, Apr 14 2015 *)
f[n_] := ((n - 2) n^n + n)/(n - 1); f[1] = 1; Array[f, 18] (* Robert G. Wilson v, Apr 15 2015 *)

a(n) = A000312(n) - A023037(n) + 1. - Michel Marcus, Apr 14 2015

A191690(n)+1. - Robert G. Wilson v, Apr 16 2015

A344431 a(n) = Sum_{k=1..n} mu(k) * n^(k - 1).

Original entry on oeis.org

1, -1, -11, -19, -654, 6439, -103298, -233543, -479042, 999089891, -23581102192, -56760341915, -23425343599215, 736752260240319, 31008836538146011, 76278687880642289, -48483493990740616866, -121045054967668089989, -104414952870932959886007

Offset: 1

Seiichi Manyama, May 19 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A008683, A023037, A127513, A292779, A343425.

a[n_] := Sum[MoebiusMu[k] * n^(k-1), {k,1,n}]; Array[a, 20] (* Amiram Eldar, May 19 2021 *)

PARI
```
a(n) = sum(k=1, n, moebius(k)*n^(k-1));
```

A119758 Numerator of Sum_{k=1..n} k^n/n^k.

Original entry on oeis.org

1, 3, 20, 225, 3789, 89341, 2821552, 115377921, 5939637425, 375840753541, 28641787322796, 2583828842108449, 271949027324094925, 32986652806128680205, 4563200871898056653504, 713455071424061222336513

Offset: 1

Alexander Adamchuk, Jun 18 2006, Jun 25 2006

a(p-1) is divisible by prime p>2. a(p) is divisible by ((p+1)/2)^2 for prime p>2.

Denominator of Sum[k^n/n^k,{k,1,n}] is equal to n^(n-1) = A000169(n). - Alexander Adamchuk, Jun 27 2006

Cf. A023037, A031971.

Cf. A000169.

Table[Numerator[Sum[k^n/n^k,{k,1,n}]],{n,1,20}]
Table[Sum[k^n/n^k,{k,1,n}]*n^(n-1),{n,1,50}] (* Alexander Adamchuk, Jun 27 2006 *)

a(n) = numerator(prod(k=2, n, 1-1/(prime(k)-1)^2)); \\ Michel Marcus, May 31 2022

a(n) = numerator(Sum_{k=1..n} k^n/n^k).
a(n) = n^(n-1)*(Sum_{k=1..n} k^n/n^k). - Alexander Adamchuk, Jun 27 2006
a(2m) is divisible by 2m+1 for integer m>0. a(2m-1) is divisible by m^2 for integer m>0. - Alexander Adamchuk, Jun 27 2006

A125571 Least prime factor of Sum_{k=0..n-1} n^k.

Original entry on oeis.org

3, 13, 5, 11, 7, 29, 3, 7, 11, 15797, 5, 53, 3, 11, 17, 10949, 7, 109912203092239643840221, 3, 43, 23, 461, 5, 11, 3, 109, 5, 59, 7, 568972471024107865287021434301977158534824481, 3, 67, 5, 31, 13, 149, 3, 7, 11, 83, 13, 173, 3, 19, 47

Offset: 2

Axel Harvey, Jan 02 2007

nonn

The sequence of largest prime factors of numbers generated by the same sum is probably identical to sequence A006486, since (n^n - 1)/(1 + n^2 + ... + n^(n-1)) = n-1.

			The sum 1 + 4 + 4^2 + 4^3 = 85 = 5 * 17 so the third term is 5.

Chai Wah Wu, Table of n, a(n) for n = 2..178

Cf. A006486.

Least prime factors of A023037.

Table[FactorInteger[Sum[n^k,{k,0,n-1}]][[1,1]],{n,2,46}] (* James C. McMahon, Dec 18 2024 *)

a(n) = factor(sum(k=0, n-1, n^k))[1, 1]; \\ Michel Marcus, Aug 20 2013

More terms from Michel Marcus, Aug 20 2013

cp's OEIS Frontend

A294316 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where A(n,k) is the sum of products of terms in all partitions of k*n into powers of k.

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A331672 Sum of all base-n numbers with digit sum n and length at most n.

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Maple

Formula

A332653 a(n) = (1/n) * Sum_{k=1..n} n^(k/gcd(n, k)).

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Magma

Mathematica

Formula

A368534 a(n) = Sum_{k=1..n} binomial(k+1,2) * n^(n-k).

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Mathematica

PARI

Formula

A068742 Largest power of n! which divides (n^n)!.

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PARI

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A077385 Triangle read by rows in which n-th row contains 2n-1 terms starting from n^0 to n^(n-1) in increasing order and then in decreasing order to n^0.

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Magma

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SageMath

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A117667 a(n) = n^n-n^(n-1)-n^(n-2)-n^(n-3)-...-n^3-n^2-n.

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Maple

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A344431 a(n) = Sum_{k=1..n} mu(k) * n^(k - 1).

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Mathematica