A062970 -id:A062970 - 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

A128981 Numbers k such that k divides Sum_{j=1..k} j^j = A001923(k).

Original entry on oeis.org

1, 4, 17, 19, 148, 1577, 3564, 4388, 5873, 6639, 8579, 62500, 376636, 792949, 996044, 1174065, 3333551, 5179004, 7516003

Offset: 1

Alexander Adamchuk, Apr 29 2007

a(20) > 10^7. - Hiroaki Yamanouchi, Aug 25 2015

Cf. A062970, A001923.

a:=0:
for n from 1 to 2000 do
    a:=a+n^n:
    if a mod n=0 then
        print(n);
    fi;
od: # Revised program from R. J. Mathar, Jun 18 2015

f=0; Do[ f=f+k^k; If[ IntegerQ[f/k], Print[k] ], {k,1,6639} ]

for(n=1,10^4, s=sum(i=1,n,Mod(i,n)^i); if(!Mod(s,n), print1(n,", "))) \\ Derek Orr, Jun 18 2015

from itertools import accumulate, count, islice
def A128981_gen(): # generator of terms
    yield 1
    for i, j in enumerate(accumulate(k**k for k in count(1)),start=2):
        if j % i == 0:
            yield i
A128981_list = list(islice(A128981_gen(),10)) # Chai Wah Wu, Jun 18 2022

a(11) and a(12) from Jon E. Schoenfield, May 09 2007
a(13) = 376636 from Alexander Adamchuk, May 03 2010
a(14)-a(16) from Lars Blomberg, May 10 2011
a(17) from Giovanni Resta, Jul 13 2015
a(18)-a(19) from Hiroaki Yamanouchi, Aug 25 2015

A349886 a(n) = Sum_{k=0..n} k^(k*n).

Original entry on oeis.org

1, 2, 18, 19749, 4295498995, 298024323402930834, 10314425729813391637014599924, 256923578002288684397369021397408936103993, 6277101735598268377660667072561845282166297358613176925573

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

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

Cf. A031971, A062970, A249459.

Table[1 + Sum[k^(k*n), {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Dec 04 2021 *)
a[n_] := Sum[If[k == 0, 1, k^(k*n)], {k, 0, n}]; Array[a, 9, 0] (* Amiram Eldar, Dec 04 2021 *)

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

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, k^k^2*x^k/(1-k^k*x)))

G.f.: Sum_{k>=0} k^(k^2) * x^k/(1 - k^k * x).

a(n) ~ n^(n^2). - Vaclav Kotesovec, Dec 04 2021

A073826 Primes of the form Sum_{k=1..n} k^k, i.e., primes in A001923.

Original entry on oeis.org

5, 3413, 50069, 10405071317, 208492413443704093346554910065262730566475781

Offset: 1

Rick L. Shepherd, Aug 13 2002

nonn

a(3) = A001923(10) = 10405071317 and the 45-digit a(4) = A001923(30) have been certified prime with Primo. Any additional terms are too big to include here.

The next term would have over 20000 digits; see A073825 for more information and updates.

			a(1) = 5 = 1^1 + 2^2 is the smallest prime of the form A001923(n) = sum_{k=1..n} k^k, namely for n = 2 = A073825(1).
a(2) = sum_{k=1..A073825(2)} k^k = 1^1 + 2^2 + 3^3 + 4^4 + 5^5 = 3413, a prime, so 3413 is in this sequence (A073825(2) = 5).

Cf. A073825 (corresponding n), A001923 (sum_{k=1..n} k^k).

Cf. A122166 (indices of primes in A062970 (sum_{k=0..n} k^k)).

Select[s=0;Table[s+=n^n,{n,5!}],PrimeQ[ # ]&] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

s=0; for(k=1,1320, s=s+k^k; if(isprime(s), print1(s,",")))

a(j) = A001923(A073825(j)) = sum_{k=1..A073825(j)} k^k.

Intersection of A001923 with A000040.

Edited by M. F. Hasler, Mar 22 2008

Typo in comment corrected by Jonathan Vos Post, Mar 23 2008

A188775 Numbers k such that Sum_{j=1..k} j^j == -1 (mod k).

Original entry on oeis.org

1, 2, 3, 6, 14, 42, 46, 1806, 2185, 4758, 5266, 10895, 24342, 26495, 44063, 52793, 381826, 543026, 547311, 805002

Offset: 1

José María Grau Ribas, Apr 10 2011

Numbers k such that A001923(k) == -1 (mod k).
a(21) > 10^7. - Hiroaki Yamanouchi, Aug 25 2015
Numbers k such that k divides A062970(k). - Jianing Song, Feb 03 2019

			6 is a term because 1^1 + 2^2 + 3^3 + 4^4 + 5^5 + 6^6 = 50069 and 50069 + 1 = 6 * 8345. - _Bernard Schott_, Feb 03 2019

Cf. A128981 (sum == 0 (mod n)), A188776 (sum == 1 (mod n)).
Cf. A057245.
Cf. A001923, A062970.

isA188775 := proc(n) add( modp(k &^ k,n),k=1..n) ; if modp(%,n) = n-1 then true; else false; end if; end proc:
for n from 1 do if isA188775(n) then printf("%d\n",n) ; end if; end do: # R. J. Mathar, Apr 10 2011

Union@Table[If[Mod[Sum[PowerMod[i,i,n],{i,1,n}],n]==n-1,Print[n];n],{n,1,10000}]

f(n)=lift(sum(k=1,n,Mod(k,n)^k));
for(n=1,10^6,if(f(n)==n-1,print1(n,", "))) \\ Joerg Arndt, Apr 10 2011

m=0;for(n=1,1000,m=m+n^n;if((m+1)%n==0,print1(n,", "))) \\ Jinyuan Wang, Feb 04 2019

sum = 0
for n in range(10000):
    sum += n**n
    if sum % (n+1) == 0:
        print(n+1, end=',')
# Alex Ratushnyak, May 13 2013

a(12)-a(16) from Joerg Arndt, Apr 10 2011

a(17)-a(20) from Lars Blomberg, May 10 2011

A349962 a(n) = Sum_{k=0..n} (2*k)^k.

Original entry on oeis.org

1, 3, 19, 235, 4331, 104331, 3090315, 108503819, 4403471115, 202762761483, 10442762761483, 594761064172811, 37115108500229387, 2518267981703965963, 184577387811646500107, 14533484387811646500107, 1223459304002440821206283, 109651494909968373175414027

Offset: 0

Seiichi Manyama, Dec 07 2021

Partial sums of A062971.

Seiichi Manyama, Table of n, a(n) for n = 0..351

Cf. A062970, A062971, A349961, A349963, A349970.

a[n_] := Sum[If[k == 0, 1, (2*k)^k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 07 2021 *)

PARI
```
a(n) = sum(k=0, n, (2*k)^k);
```

a(n) ~ 2^n * n^n. - Vaclav Kotesovec, Dec 07 2021

A108397 Sums of rows of the triangle in A108396.

Original entry on oeis.org

0, 2, 10, 66, 692, 9780, 167982, 3362828, 76695880, 1961316270, 55555555610, 1726135607262, 58359930206844, 2132745542253872, 83767436069591302, 3518790190560477240, 157412216095654840592, 7471013615160978901626

Offset: 0

Reinhard Zumkeller, Jun 02 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A007778, A062970.

a108397 0 = 0
a108397 1 = 2
a108397 n = n * (n^(n+1) + n^2 - 2) `div` (2 * (n-1))
-- Reinhard Zumkeller, Mar 31 2015

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

A326501 a(n) = Sum_{k=0..n} (-k)^k.

Original entry on oeis.org

1, 0, 4, -23, 233, -2892, 43764, -779779, 15997437, -371423052, 9628576948, -275683093663, 8640417354593, -294234689237660, 10817772136320356, -427076118244539019, 18019667955465012597, -809220593930871751580, 38537187481365665823844

Offset: 0

Seiichi Manyama, Sep 12 2019

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A001099, A062970, A177885.

a:= proc(n) option remember; `if`(n<0, 0, (-n)^n+a(n-1)) end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 2019

RecurrenceTable[{a[0] == 1, a[n] == a[n-1] + (-n)^n}, a, {n, 0, 23}] (* Jean-François Alcover, Nov 27 2020 *)

PARI
```
{a(n) = sum(k=0, n, (-k)^k)}
```

from itertools import accumulate, count, islice
def A326501_gen(): # generator of terms
    yield from accumulate((-k)**k for k in count(0))
A326501_list = list(islice(A326501_gen(),10)) # Chai Wah Wu, Jun 18 2022

a(n) = 1 + (-1)^n * A001099(n).

A350008 a(n) = Sum_{k=0..n} k^(2*k).

Original entry on oeis.org

1, 2, 18, 747, 66283, 9831908, 2186614244, 680409687093, 282155386397749, 150376790683396870, 100150376790683396870, 81502899763630444510191, 79578350103154474577951727, 91812908543371771132977567736

Offset: 0

Seiichi Manyama, Dec 08 2021

nonn

Partial sums of A062206.

Seiichi Manyama, Table of n, a(n) for n = 0..214

Cf. A000330, A062206, A062970, A100262, A249459, A349962.

a[n_] := Sum[If[k == 0, 1, k^(2*k)], {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Dec 08 2021 *)

PARI
```
a(n) = sum(k=0, n, k^(2*k));
```

a(n) ~ n^(2*n). - Vaclav Kotesovec, Dec 08 2021

A353009 a(n) = Sum_{k=0..floor(n/2)} (n-2k)^(n-2k).

Original entry on oeis.org

1, 1, 5, 28, 261, 3153, 46917, 826696, 16824133, 388247185, 10016824133, 285699917796, 8926117272389, 303160806510049, 11120932942830405, 438197051187369424, 18457865006652382021, 827678458937524133601, 39364865940303189957445

Offset: 0

Seiichi Manyama, Apr 16 2022

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A062970, A353018.

Cf. A061787, A061788, A352082, A353013.

a[n_] := Sum[If[2*k == n, 1, (n - 2*k)^(n - 2*k)], {k, 0, Floor[n/2]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 16 2022 *)

PARI
```
a(n) = sum(k=0, n\2, (n-2*k)^(n-2*k));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k)/(1-x^2))

G.f.: ( Sum_{k>=0} (k * x)^k )/(1 - x^2).

a(2*n-1) = A061787(n), a(2*n) = A061788(n) + 1. - Seiichi Manyama, Apr 17 2022

cp's OEIS Frontend

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

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A128981 Numbers k such that k divides Sum_{j=1..k} j^j = A001923(k).

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A349886 a(n) = Sum_{k=0..n} k^(k*n).

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A073826 Primes of the form Sum_{k=1..n} k^k, i.e., primes in A001923.

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A188775 Numbers k such that Sum_{j=1..k} j^j == -1 (mod k).

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A349962 a(n) = Sum_{k=0..n} (2*k)^k.

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A108397 Sums of rows of the triangle in A108396.

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A353009 a(n) = Sum_{k=0..floor(n/2)} (n-2k)^(n-2k).