A001923 -id:A001923 - cp's OEIS frontend

A117887 Number of labeled trees on <= n nodes.

1, 4, 20, 145, 1441, 18248, 280392, 5063361, 105063361, 2463011052, 64380375276, 1856540769313, 58550453144609, 2004745521503984, 74062339559431920, 2936485391069247713, 124376016487663499489, 5604762874272465685428

Offset: 1

Jonathan Vos Post, May 03 2006

A000178 = Sum_{k=1..n} k^(k-1). A001923 = Sum_{k=1..n} k^k. A061789 = Sum_{k=1..n} p(k)^p(k), p(k) = k-th prime. a(n) = number of spanning trees in complete graphs K_i on i <= n labeled nodes. Also is partial sum of counts of parking functions, noncrossing partitions, critical configurations of the chip firing game, allowable pairs sorted by a priority queue. a(14) = 58550453144609 is prime.

Cf. A000178, A000272, A001923, A061789.

a:=n->sum ((j+2)^j, j=0..n): seq(a(n), n=0..17); # Zerinvary Lajos, Dec 17 2008

Table[Sum[k^(k-2),{k,2,n}],{n,2,25}] (* Harvey P. Dale, Jul 11 2011 *)

a(n) = Sum_{k=2..n} k^(k-2). a(n) = Sum_{k=2..n} A000272(k).

A185353 a(n) = (1^1 + 2^2 . . . + n^n) mod 10.

Original entry on oeis.org

1, 5, 2, 8, 3, 9, 2, 8, 7, 7, 8, 4, 7, 3, 8, 4, 1, 5, 4, 4, 5, 9, 6, 2, 7, 3, 6, 2, 1, 1, 2, 8, 1, 7, 2, 8, 5, 9, 8, 8, 9, 3, 0, 6, 1, 7, 0, 6, 5, 5, 6, 2, 5, 1, 6, 2, 9, 3, 2, 2, 3, 7, 4, 0, 5, 1, 4, 0, 9, 9, 0, 6, 9, 5, 0, 6, 3, 7, 6, 6, 7, 1, 8, 4, 9, 5, 8, 4, 3, 3, 4, 0, 3, 9, 4, 0, 7, 1, 0, 0

Offset: 1

Muhammed Hedayet, Jan 26 2012

nonn

The last digit of 1^1 + 2^2 +...+ n^n, which has period 100.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000312 (n^n), A001923 (1^1 + 2^2 +...+ n^n).

Mod[Accumulate[Table[n^n, {n, 200}]], 10] (* T. D. Noe, Jan 27 2012 *)

a(n) = sum(k=1, n, k^k) % 10; \\ Michel Marcus, Jun 28 2017

from itertools import accumulate, count, islice
def A185353_gen(): # generator of terms
    yield from accumulate((pow(k,k,10) for k in count(1)),func=lambda x,y:(x+y)%10)
A185353_list = list(islice(A185353_gen(),40)) # Chai Wah Wu, Jun 17 2022

Extended by T. D. Noe, Jan 27 2012

A062814 a(n) = Sum_{i=0..n-1} i * (n - i)^(n - i).

Original entry on oeis.org

0, 1, 6, 38, 326, 3739, 53808, 927420, 18578248, 423649565, 10828720882, 306545462810, 9518362652994, 321605286435431, 11745699035775884, 461063683165975712, 19357125741005727156, 865493449685182242777

Offset: 1

Olivier Gérard, Jun 23 2001

Partial sums of A001923. - Amiram Eldar, Mar 26 2022

Seiichi Manyama, Table of n, a(n) for n = 1..387
Andrew Cusumano, Problem 11724, The American Mathematical Monthly, Vol. 120, No. 7 (2013), p. 661; A Limit Computation, Solution to Problem 11724 by Hosam M. Mahmoud, ibid., Vol. 122, No. 7 (2015), pp. 706-707.

Cf. A001113, A001923, A062813.

Table[Sum[i*(n - i)^(n - i), {i, 0, -1 + n}], {n, 1, 18}]

a(n) = sum(i=0, n-1, i*(n-i)^(n-i)); \\ Michel Marcus, Mar 26 2019

From Amiram Eldar, Mar 26 2022: (Start)
a(n) = Sum_{k=0..n-1} A001923(k).
Limit_{n->oo} a(n+2)/a(n+1) - a(n+1)/a(n) = e (Cusumano, 2013). (End)

A065980 Inverse binomial transform of [1^1,2^2,3^3,...], shifted right by one index.

Original entry on oeis.org

1, 3, 20, 186, 2248, 33340, 585744, 11891236, 273854368, 7053523236, 200894140120, 6268924259884, 212691682554960, 7795165961244532, 306908654169113416, 12918649608270463740, 578931362074039774144

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 09 2001

{0, a(n),n=1,...} = inverse binomial transform of {A001923(m), m=0,...} [From Tilman Neumann, Dec 17 2008]

Vincenzo Librandi, Table of n, a(n) for n = 1..200
F. Ellermann, Illustration of binomial transforms
N. J. A. Sloane, Transforms

Cf. A001923, A069856.

CoefficientList[Series[-E^(-x)*LambertW[-x]/(1+LambertW[-x])^3/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 17 2014 *)

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

O.g.f.: Sum_{n>0} (n*x/(1+x))^n. E.g.f.: int(-exp(-x)*LambertW(-x)/(1+LambertW(-x))^3/x, x). - Vladeta Jovovic, Apr 12 2003

a(n) ~ n^n * exp(-exp(-1)). - Vaclav Kotesovec, Feb 17 2014

A109392 Partial sums of A109391.

Original entry on oeis.org

0, 1, 13, 175, 2735, 49610, 1029386, 24088590, 628068366, 18061990371, 568061990371, 19398632250697, 714854467214665, 28276489167109688, 1195037205850701368, 53742304051553826368, 2562499498076052846144

Offset: 0

Rick L. Shepherd, Jun 27 2005

nonn

The sum of all the terms of all A001923(n) sequences having up to n terms all chosen from {1,2,...,n}.

Cf. A109391, A001923 (sum k^k, k=1..n).

A117897 Number of labeled trees on prime numbers of nodes through n-th prime.

Original entry on oeis.org

1, 4, 129, 16936, 2357964627, 1794518358664, 2862424846028174457, 5483249282630830360396, 39471589603944768518079950019, 3053134546009996125349281528007992109928

Offset: 1

Jonathan Vos Post, May 03 2006

A000178 = Sum_{k=1..n} k^(k-1). A001923 = Sum_{k=1..n} k^k. A061789 = Sum_{k=1..n} prime(k)^prime(k), prime(k) = k-th prime.

First differences a(n+1) - a(n) for n=1,...,9 are A076931(j) at j=3, 5, 7, 11, 13, 17, 19, 23 and 29. - R. J. Mathar, May 01 2007

			a(1) = number of labeled trees on prime(1) numbers of nodes = number of labeled trees on 2 nodes = A000272(2) = 2^0 = 1.
a(2) = number of labeled trees on prime(1) or prime(2) numbers of nodes = number of labeled trees on 2 or 3 nodes = A000272(2)+A000272(3) = 2^0 + 3^1 = 4.
a(3) = number of labeled trees on prime(1) or prime(2) or prime(3) numbers of nodes = number of labeled trees on 2 or 3 or 5 nodes = A000272(2)+A000272(3)+A000272(5) = 2^0 + 3^1 + 5^3 = 129.

G. C. Greubel, Table of n, a(n) for n = 1..75

Cf. A000040, A000178, A000272, A001923, A061789.

Table[Sum[Prime[k]^(Prime[k] -2), {k,n}], {n,20}] (* G. C. Greubel, Sep 27 2021 *)

[sum( nth_prime(k)^(nth_prime(k) -2) for k in (1..n)) for n in (1..20)] # G. C. Greubel, Sep 27 2021

a(n) = Sum_{k=1..n} prime(k)^(prime(k)-2).

a(n) = Sum_{k=1..n} A000272(A000040(k)).

A120929 Partial sums of n^(n^2), A002489.

Original entry on oeis.org

1, 2, 18, 19701, 4294986997, 298023228171940122, 10314424798788558774343889178, 256923577521069192513410265783009965210785, 6277101735386681020759366944276858929512621227473999723681

Offset: 0

Jonathan Vos Post, Aug 18 2006

After 2, can this ever be prime? This is to A001923 Sum k^k, k=1..n, as k^k^k is to k^k.

			a(0) = 1 because A002489(0) is given formally as 0^0^0 = 1.
a(1) = 2 because 1 + (1^1)^1 = 1 + 1 = 2.
a(2) = 18 because 2 + (2^2)^2 = 2 + 16 = 18.
a(3) = 19701 because 18 + (3^3)^3 = 18 + 19683 = 19701.
a(4) = 4294986997 = 19701 + (4^4)^4 = 19701 + 4294967296.

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A001923, A002489, A002488, A001329, A002488, A023813, A076113, A090588.

Accumulate[Join[{1},Table[n^(n^2),{n,9}]]] (* Harvey P. Dale, Apr 10 2014 *)

a(n) = Sum_{i=0..n} i^(i^2). a(n) = Sum_{i=0..n} (i^i)^i. In this sequence, we formally define 0^0 = 1.

More terms from Harvey P. Dale, Apr 10 2014

A175232 The smallest prime divisor of 1 + 2^2 + 3^3 + ... + n^n.

Original entry on oeis.org

5, 2, 2, 3413, 50069, 2, 2, 7, 10405071317, 2, 2, 88799, 3, 2, 2, 3, 3, 2, 2, 5, 3, 2, 2, 3, 7, 2, 2, 7, 208492413443704093346554910065262730566475781, 2, 2, 3, 17, 2, 2, 5, 61, 2, 2, 71, 11, 2, 2, 11, 7, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 23, 3, 2, 2, 3, 44818693, 2, 2, 5

Offset: 2

Michel Lagneau, Mar 09 2010

nonn

			a(2) = 5 divides 1 + 2^2.
a(3) = 2 divides 1 + 2^2 + 3^3 = 32.
a(4) = 2 divides 1 + 2^2 + 3^3 + 4^4 = 288.
a(5) = 3413 divides 1 + 2^2 + 3^3 + 4^4 + 5^5 = 3413.
a(13) = 88799 divides 1 + 2^2 + 3^3 + ... + 13^13 = 88799 * 3514531963.

Cf. A073826, A122166.

with(numtheory): s :=1: for n from 2 to 60 do ;s := s+ n^n: s1 := ifactors(s)[2] : s2 :=s1[i][1], i=1..nops(s1):print(s1[1][1]):od:

a[n_] := FactorInteger[Sum[k^k, {k, 1, n}]][[1, 1]]; Array[a, 20, 2] (* Amiram Eldar, Feb 04 2020 *)

a(n) = A020639(A001923(n)).

Edited by R. J. Mathar, Mar 16 2010

a(61)-a(65) from Amiram Eldar, Feb 04 2020

A214662 Greatest prime divisor of 1 + 2^2 + 3^3 + ... + n^n.

Original entry on oeis.org

5, 2, 3, 3413, 50069, 8089, 487, 2099, 10405071317, 1274641129, 164496735539, 3514531963, 15624709, 23747111, 10343539, 56429700667, 1931869473647715169, 2383792821710269, 144326697012150473, 2053857208873393249, 128801386946535261205906957, 2298815880166789

Offset: 2

Michel Lagneau, Jul 24 2012

nonn

			a(2) = 5 divides 1 + 2^2 ;
a(3) = 2 divides 1 + 2^2 + 3^3 = 32 ;
a(4) = 3 divides 1 + 2^2 + 3^3 + 4^4 = 288 = 2^5*3^2 ;
a(5) = 3413 divides 1 + 2^2 + 3^3 + 4^4 + 5^5 = 3413.
a(13) = 3514531963 divides 1 + 2^2 + 3^3 + ... + 13^13 = 88799 * 3514531963.

Amiram Eldar, Table of n, a(n) for n = 2..73

Cf. A001923, A006530, A073826, A122166, A175232.

[Max(PrimeDivisors(&+[k^k:k in [1..n]])):n in [2..23]]; // Marius A. Burtea, Feb 09 2020

with (numtheory):
s:= proc(n) option remember; `if`(n=1, 1, s(n-1)+n^n) end:
a:= n-> max(factorset(s(n))[]):
seq (a(n), n=2..23);  # Alois P. Heinz, Jul 24 2012

s = 1; Table[s = s + n^n; FactorInteger[s][[-1, 1]], {n, 2, 24}] (* T. D. Noe, Jul 25 2012 *)
Module[{nn=30,lst},lst=Table[n^n,{n,nn}];Table[FactorInteger[Total[Take[lst,k]]][[-1,1]],{k,2,nn}]] (* Harvey P. Dale, Oct 09 2022 *)

a(n) = vecmax(factor(sum(k=1, n, k^k))[,1]); \\ Michel Marcus, Feb 09 2020

a(n) = A006530(A001923(n)).

A343932 a(n) = (Sum_{k=1..n} k^k) mod n.

Original entry on oeis.org

0, 1, 2, 0, 3, 5, 5, 4, 1, 7, 3, 4, 11, 13, 3, 4, 0, 15, 0, 4, 14, 13, 10, 20, 22, 11, 25, 20, 21, 1, 18, 4, 6, 17, 27, 12, 31, 27, 20, 28, 6, 41, 34, 32, 31, 45, 45, 4, 11, 25, 39, 48, 21, 45, 46, 12, 53, 47, 41, 32, 9, 5, 55, 4, 25, 7, 47, 8, 45, 19, 12, 60, 50, 43, 20, 60, 54, 29, 72, 36, 70, 31, 74, 40, 69, 7, 18, 20, 63, 3, 24, 32

Offset: 1

Seiichi Manyama, May 04 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001923, A128981, A182398, A188775, A188776.

a[n_] := Mod[Sum[PowerMod[k, k, n], {k, 1, n}], n]; Array[a, 100] (* Amiram Eldar, May 04 2021 *)

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

def A343932(n): return sum(pow(k,k,n) for k in range(1,n+1)) % n # Chai Wah Wu, Jun 18 2022

a(n) = A001923(n) mod n.

cp's OEIS Frontend

A117887 Number of labeled trees on <= n nodes.

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A185353 a(n) = (1^1 + 2^2 . . . + n^n) mod 10.

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A062814 a(n) = Sum_{i=0..n-1} i * (n - i)^(n - i).

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A065980 Inverse binomial transform of [1^1,2^2,3^3,...], shifted right by one index.

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Mathematica

PARI

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A109392 Partial sums of A109391.

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A117897 Number of labeled trees on prime numbers of nodes through n-th prime.

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Mathematica

Sage

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A120929 Partial sums of n^(n^2), A002489.

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A175232 The smallest prime divisor of 1 + 2^2 + 3^3 + ... + n^n.

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