A060946 -id:A060946 - cp's OEIS frontend

A001923 a(n) = Sum_{k=1..n} k^k.

0, 1, 5, 32, 288, 3413, 50069, 873612, 17650828, 405071317, 10405071317, 295716741928, 9211817190184, 312086923782437, 11424093749340453, 449317984130199828, 18896062057839751444, 846136323944176515621

Offset: 0

N. J. A. Sloane

Starting from the second term, 1, the terms could be described as the special case (n=1; j=1) of the following general formula: a(n) =  Sum [(n + k - 1)]^(k) n=1; j=1; i=1,2,3,...,... For (n=0; j=1) the formula yields A062815 n=0; j=1; i=2,3,4,... For (n=2; j=0) we get A060946 and for (n=3; j=0) A117887. - Alexander R. Povolotsky, Sep 01 2007
From Luan Alberto Ferreira, Aug 01 2017: (Start)
If n == 0 or 3 (mod 4), then a(n) == 0 (mod 4).
If n == 0, 4, 7, 14, 15 or 17 (mod 18), then a(n) == 0 (mod 3). (End)
Called the hypertriangular function by M. K. Azarian. - Light Ediand, Nov 19 2021

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, p. 308.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
Mohammad K. Azarian, On the hyperfactorial function, hypertriangular function and the discriminants of certain polynomials, Int. J. Pure Appl. Math., Vol. 36, No. 2 (2007), pp. 251-257.
Andrew Cusumano, Problem H-656, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; A Sequence Tending To e, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.
G. W. Wishard (proposer) and F. Underwood (solution), Problem 4155: Bound for a Finite Sum, Amer. Math. Monthly, Vol. 53, No. 8 (1946), pp. 471-473.

Cf. A073825, A062970 (another version).
Cf. A062815, A060946, A117887.
Cf. A000312, A002109.

a001923 n = a001923_list !! n
a001923_list = scanl (+) 0 $ tail a000312_list
-- Reinhard Zumkeller, Jul 11 2014

Accumulate[Join[{0},Table[k^k,{k,20}]]] (* Harvey P. Dale, Feb 11 2015 *)

for(n=1,20,print1(sum(x=1,n,x^x), ", ")) \\ Jorge Coveiro, Dec 24 2004

# generates initial segment of sequence
from itertools import accumulate
def f(k): return 0 if k == 0 else k**k
def aupton(nn): return list(accumulate(f(k) for k in range(nn+1)))
print(aupton(17)) # Michael S. Branicky, Feb 12 2022

a(n) = A062970(n) - 1.
a(n+1)/a(n) > e*n and a(n+1)/a(n) is asymptotic to e*n. - Benoit Cloitre, Sep 29 2002
For n > 0: a(n) = a(n-1) + A000312(n). - Reinhard Zumkeller, Jul 11 2014
Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - Amiram Eldar, Jan 03 2022

A209424 Triangle defined by g.f.: A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^n * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 76, 347, 76, 1, 1, 701, 20429, 20429, 701, 1, 1, 8477, 1919660, 10707908, 1919660, 8477, 1, 1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1, 1, 2223278, 47484618291, 12099129236936, 72078431500368

Offset: 0

Paul D. Hanna, Mar 08 2012

Column 1 is A060946.
Column 2 is A209425.
Row sums equal A167007.
Antidiagonal sums equal A166894.
Central terms form A209426.

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 12, 12, 1;
1, 76, 347, 76, 1;
1, 701, 20429, 20429, 701, 1;
1, 8477, 1919660, 10707908, 1919660, 8477, 1;
1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1;
1, 2223278, 47484618291, 12099129236936, 72078431500368, 12099129236936, 47484618291, 2223278, 1; ...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+12*y+12*y^2+y^3)*x^3 + (1+76*y+20429*y^2+76*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^2*y + y^2)*x^2/2
+ (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3
+ (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4
+ (1 + 5^5*y + 10^5*y^2 + 10^5*y^3 + 5^5*y^4 + y^5)*x^5/5 +...
in which the coefficients are found in triangle A209427.

Paul D. Hanna, Table of n, a(n) for n = 0..495 for Rows 0..30 of this triangle in flattened form.

Cf. A060946, A209425, A167007, A166894, A209426, A209427, A209196.

{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m,k)^m*y^k))+x*O(x^n)),n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

A345094 a(n) = Sum_{k=1..n} floor(n/k)^(floor(n/k) - 1).

Original entry on oeis.org

1, 3, 11, 68, 630, 7790, 117664, 2097224, 43046801, 1000000643, 25937425245, 743008378547, 23298085130341, 793714773371879, 29192926025508929, 1152921504608944840, 48661191875668966346, 2185911559738739586562, 104127350297911284587436

Offset: 1

Seiichi Manyama, Jun 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A060946, A262843, A344551, A345030, A345098.

a[n_] := Sum[Floor[n/k]^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 08 2021 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(j=1, N, (1-x^j)*sum(k=1, N, k^(k-1)*x^(j*k)))/(1-x))

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

A355950 a(n) = Sum_{k=1..n} k^(k-1) * floor(n/k).

Original entry on oeis.org

1, 4, 14, 81, 707, 8495, 126145, 2223364, 45270095, 1045270723, 26982695325, 769991073865, 24068076196347, 817782849568143, 30010708874959403, 1182932213483903598, 49844124089150772080, 2235755683827890358557, 106363105981739131891399

Offset: 1

Seiichi Manyama, Jul 21 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Partial sums of A262843.
Cf. A006218, A268235, A344814, A344815, A344816.
Cf. A060946, A355887.

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

a(n) = sum(k=1, n, sumdiv(k, d, d^(d-1)));

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k-1)*x^k/(1-x^k))/(1-x))

def A355950(n): return n*(1+n**(n-2))+sum(k**(k-1)*(n//k) for k in range(2,n)) if n>1 else 1 # Chai Wah Wu, Jul 21 2022

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

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

A228899 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^(k+1) * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 12, 1, 1, 10, 71, 76, 1, 1, 15, 281, 2153, 701, 1, 1, 21, 861, 29166, 129509, 8477, 1, 1, 28, 2212, 244725, 7664343, 12391414, 126126, 1, 1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1, 1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1

Offset: 0

Paul D. Hanna, Sep 07 2013

Note that the following g.f. does NOT yield an integer triangle: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^k * y^k ).

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 6, 12, 1;
1, 10, 71, 76, 1;
1, 15, 281, 2153, 701, 1;
1, 21, 861, 29166, 129509, 8477, 1;
1, 28, 2212, 244725, 7664343, 12391414, 126126, 1;
1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1;
1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1; ...
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+12*y^2+y^3)*x^3 + (1+10*y+71*y^2+76*y^3+y^4)*x^4 + (1+15*y+281*y^2+2153*y^3+701*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x)*x
+ (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^2*y + 3^3*y^2 + y^3)*x^3/3
+ (1+ 4^2*y + 6^3*y^2 + 4^4*y^3 + x^4)*x^4/4
+ (1+ 5^2*y + 10^3*y^2 + 10^4*y^3 + 5^5*y^4 + y^5)*x^5/5
+ (1+ 6^2*y + 15^3*y^2 + 20^4*y^3 + 15^5*y^4 + 6^6*y^5 + y^6)*x^6/6 +...
in which the coefficients form A219207(n,k) = binomial(n, k)^(k+1).

Cf. A184730 (row sums), A181070 (antidiagonal sums), A060946 (diagonal).

Cf. related triangles: A219207, A209424, A228904.

{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m, j)^(j+1)*y^j))+x*O(x^n)), n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A232821 a(n) = n^(n-1) - Sum_{k=1..n-1} k^(k-1).

Original entry on oeis.org

1, 1, 6, 52, 549, 7075, 109172, 1971026, 40823443, 954730001, 24892154602, 716025676088, 22528094057193, 769646697066375, 28375143175948712, 1122910795732014438, 47478259662185188967, 2136067435649547983973, 101891594614083396452878

Offset: 1

Derek Orr, Nov 30 2013

			6^5 - 5^4 - 4^3 - 3^2 - 2^1 - 1^0 = 7075 so a(6) = 7075.

Derek Orr, Table of n, a(n) for n = 1..100

Cf. A060946, A000169.

a[n_] := n^(n - 1) - Sum[i^(i - 1), {i, 1, n - 1}]; Table[a[n], {n, 20}] (* Carlos Eduardo Olivieri, May 29 2015 *)

vector(20,n,n^(n-1)-sum(i=1,n-1,i^(i-1))) \\ Derek Orr, Apr 05 2015

def sub(n):
  num = n**(n-1)
  for i in range(0, n-1):
      num -= (i+1)**i
  return num
n = 1
while n < 100:
  print(sub(n), end=', ')
  n += 1

a(n) = n^(n-1) - A060946(n-1).

a(n) = A000169(n) - Sum_{k=1..n-1} A000169(k).

Name edited by Derek Orr, Apr 05 2015

A128884 Sum of all matrix elements of n X n Vandermonde matrix of numbers 1,2,...,n, i.e., the matrix A with A[i,j] = i^(j-1), 1 <= i <= n, 1 <= j <= n.

Original entry on oeis.org

1, 5, 23, 144, 1279, 15035, 219463, 3816512, 76928685, 1762344781, 45207853767, 1283438430208, 39944988007339, 1352308628695895, 49471532968242991, 1944732944768690432, 81748776383970349721, 3659142661552743151353

Offset: 1

Alexander Adamchuk, Apr 18 2007

nonn

p divides a(p+1) for odd primes p.
p^2 divides a(p+1) for prime p = {3, 7, 71, ...}.
Determinant of n X n Vandermonde matrix of numbers 1,2,...,n equals Product_{k=1..n-1} k! = A000178(n-1) (Superfactorials).

Eric Weisstein's World of Mathematics, Vandermonde Matrix

Cf. A060946 = Trace of Vandermonde matrix of numbers 1, 2, ..., n.

Cf. A000178 = Superfactorials: product of first n factorials.

Table[ n + Sum[ (i^n-1)/(i-1), {i,2,n} ], {n,1,25} ]

a(n) = Sum_{i=1..n, j=1..n} i^(j-1).

a(n) = n + Sum_{i=2..n} (i^n-1)/(i-1).

A276455 Primes of the form Sum_{k=1..n} k^(k-1).

Original entry on oeis.org

3, 701, 45269999

Offset: 1

Robert C. Lyons, Sep 06 2016

Searched up to n = 5000.
a(4) has 38019 digits (1973212031 ... 7493445627) and corresponds to n=9553. - Robert Price, Sep 23 2016; [number of digits in a(4) corrected by Jon E. Schoenfield, Nov 06 2016]
No other primes corresponding to n < 80000. - Robert Price, Mar 17 2017

			3 is in the sequence because 3 is prime and 3 = 2^1 + 1^0.
701 is in the sequence because 701 is prime and 701 = 5^4 + 4^3 + 3^2 + 2^1 + 1^0.
45269999 is in the sequence because 45269999 is prime and 45269999 = 9^8 + 8^7 + 7^6 + 6^5 + 5^4 + 4^3 + 3^2 + 2^1 + 1^0.

Sebastiao Antonio da Silva, Prime Curios: 9^8 + 8^7 + 7^6 + 6^5 + 5^4 + 4^3 + 3^2 + 2^1 + 1^0 is prime

Primes in A060946.

Select[Accumulate[Table[n^(n-1),{n,100}]],PrimeQ] (* Harvey P. Dale, Apr 13 2020 *)

sum = 0
seq = []
max_n = 2500
for n in range(1, max_n+1):
    sum += n^(n-1)
    if is_prime(sum):
        seq.append(n)
print(seq)

A349928 a(n) = Sum_{k=0..n} (k+n)^k.

Original entry on oeis.org

1, 3, 20, 246, 4481, 107129, 3157836, 110504876, 4473749677, 205615442135, 10574135574388, 601527803412298, 37500537926181449, 2542321872054610333, 186209553386691383388, 14653121207168215024624, 1232879877057607865696085, 110444572988776439826640683

Offset: 0

Seiichi Manyama, Dec 05 2021

nonn

Cf. A026898, A031973, A050409, A060946, A062970, A074209, A116149, A117887.

a[0] = 1; a[n_] := Sum[(k + n)^k, {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 05 2021 *)

PARI
```
a(n) = sum(k=0, n, (k+n)^k);
```

a(n) ~ 2^n * n^n. - Vaclav Kotesovec, Dec 06 2021

cp's OEIS Frontend

A001923 a(n) = Sum_{k=1..n} k^k.

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A209424 Triangle defined by g.f.: A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^n * y^k ), as read by rows.

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A345094 a(n) = Sum_{k=1..n} floor(n/k)^(floor(n/k) - 1).

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A355950 a(n) = Sum_{k=1..n} k^(k-1) * floor(n/k).

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A228899 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^(k+1) * y^k ), as read by rows.

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A232821 a(n) = n^(n-1) - Sum_{k=1..n-1} k^(k-1).

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A128884 Sum of all matrix elements of n X n Vandermonde matrix of numbers 1,2,...,n, i.e., the matrix A with A[i,j] = i^(j-1), 1 <= i <= n, 1 <= j <= n.

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