A001923 -id:A001923 - cp's OEIS frontend

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

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

A188776 Numbers n such that Sum_{k=1..n} k^k == 1 (mod n).

Original entry on oeis.org

1, 2, 9, 30, 6871, 185779, 208541, 813162, 864355, 2573155

Offset: 1

José María Grau Ribas, Apr 10 2011

Numbers n such that A001923(n) == 1 (mod n).

a(11) > 10^7. - Hiroaki Yamanouchi, Aug 25 2015

Cf. A001923, A128981 (sum == 0 mod n), A188775 (sum == -1 mod n).

Union@Table[If[Mod[Sum[PowerMod[i,i,n],{i,1,n}],n]==1,Print[n];n],{n,1,20000}]

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

from itertools import accumulate, count, islice
def A188776_gen(): # generator of terms
    yield 1
    for i, j in enumerate(accumulate(k**k for k in count(2)),start=2):
        if not j % i:
            yield i
A188776_list = list(islice(A188776_gen(),5)) # Chai Wah Wu, Jun 18 2022

a(6)-a(9) from Lars Blomberg, May 10 2011

a(1) inserted and a(10) added by Hiroaki Yamanouchi, Aug 25 2015

A342449 a(n) = Sum_{k=1..n} gcd(k,n)^k.

Original entry on oeis.org

1, 5, 29, 262, 3129, 46705, 823549, 16777544, 387421251, 10000003469, 285311670621, 8916100581446, 302875106592265, 11112006826387025, 437893890391180013, 18446744073743123788, 827240261886336764193, 39346408075299116257065

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Cf. A000010, A001923, A018804, A050873, A231712, A332517, A341036, A342370, A342389, A342423.

a[n_] := Sum[GCD[k, n]^k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)

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

If p is prime, a(p) = p-1 + p^p = A231712(p).

A344430 a(n) = Sum_{k=1..n} mu(k) * k^k.

Original entry on oeis.org

1, -3, -30, -30, -3155, 43501, -780042, -780042, -780042, 9999219958, -275312450653, -275312450653, -303150419042906, 10808856406515110, 448702746787374485, 448702746787374485, -826791559139549389692, -826791559139549389692

Offset: 1

Seiichi Manyama, May 19 2021

sign

Cf. A001923, A002321, A008683, A307653, A336276, A336277, A336278, A336279, A344429.

a[n_] := Sum[MoebiusMu[k] * k^k, {k,1,n}]; Array[a, 20] (* Amiram Eldar, May 19 2021 *)
Accumulate[Table[MoebiusMu[n]n^n,{n,20}]] (* Harvey P. Dale, Jan 25 2022 *)

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

from sympy import mobius
def A344430(n): return sum(mobius(k)*k**k for k in range(1,n+1)) # Chai Wah Wu, Apr 05 2023

A074209 a(n) = Sum_{i=n+1..2n} i^n.

Original entry on oeis.org

2, 25, 405, 8418, 216400, 6668779, 240361121, 9936764996, 463893277176, 24148657338925, 1387253043076813, 87185783860333910, 5951020164442347800, 438417132703015536399, 34673851743509883542625

Offset: 1

Zak Seidov, Sep 22 2002

nonn

A rapidly growing sequence. An even more rapidly growing sequence, the sum of next n terms of the form i^i, is given in A074309. Sum of first n terms of the form i^n is A031971. Sum of first n terms of the form i^i is A001923.

			a(2) = 25 = 3^2 + 4^2, a(3) = 405 = 4^3 + 5^3 + 6^3, a(4) = 8418 = 5^4 + 6^4 + 7^4 + 8^4, a(5) = 216400 = 6^4 + 7^5 + 8^5 + 9^5 + 10^5.

Seiichi Manyama, Table of n, a(n) for n = 1..351

Cf. A001923, A031971, A074309.

Mathematica
```
Table[Sum[i^n, {i, n+1, 2n}], {n, 20}]
```

a(n) = sum(k=n+1, 2*n, k^n); \\ Seiichi Manyama, Dec 05 2021

From Wesley Ivan Hurt, Jan 28 2021: (Start)
a(n) = Sum_{k=1..n} (n+k)^n.
a(n) = Zeta(-n,n+1) - Zeta(-n,2*n+1), where Zeta is the Hurwitz zeta function. (End)
a(n) ~ (2*n)^n / (1 - exp(-1/2)). - Vaclav Kotesovec, Dec 06 2021

Name changed by Wesley Ivan Hurt, Jan 28 2021

A122166 Numbers n such that 1 + Sum k^k (k=1..n) is prime.

Original entry on oeis.org

1, 52, 124, 431

Offset: 1

Alexander Adamchuk, Aug 23 2006

nonn

Primes of the form A001923[n] = Sum k^k (k=1..n) are given in A073826[n] and their indices are given in A073825[n] = {2,5,6,10,30,...}.

Cf. A073825, A001923.

s=1; Do[s=s+k^k;If[PrimeQ[s],Print[{k,s}]],{k,1,500}]

A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 4, 4, 5, 3, 6, 5, 5, 1, 7, 5, 8, 4, 6, 6, 9, 3, 9, 7, 8, 5, 10, 6, 11, 1, 7, 8, 7, 5, 12, 9, 8, 4, 13, 7, 14, 6, 7, 10, 15, 3, 16, 10, 9, 7, 16, 9, 8, 5, 10, 11, 17, 6, 18, 12, 8, 1, 9, 8, 19, 8, 11, 8, 20, 5, 21, 13, 11, 9, 9, 9, 22, 4, 16, 14, 23, 7, 10, 15, 12, 6

Offset: 1

Ilya Gutkovskiy, May 05 2018

nonn

			a(72) = 5 because 72 = 2^3*3^2 = prime(1)^3*prime(2)^2 and 1^3 + 2^2 = 5.

Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A002110, A003963, A008475, A056239, A066328, A156061, A222416.

a[n_] := Plus @@ (PrimePi[#[[1]]]^#[[2]]& /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 88}]

If gcd(u,v) = 1 then a(u*v) = a(u) + a(v).
a(p^k) = A000720(p)^k where p is a prime.
a(A002110(m)^k) = 1^k + 2^k + ... + m^k.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A038580(k)) = A006450(k).
a(A001248(k)) = a(A011757(k)) = A000290(k).
a(A030078(k)) = a(A055875(k)) = A000578(k).
a(A002110(k)) = a(A011756(k)) = A000217(k).
a(A061742(k)) = A000330(k).
a(A115964(k)) = A000537(k).
a(A080696(k)) = A007504(k).
a(A076954(k)) = A001923(k).

A345176 a(n) = Sum_{k=1..n} floor(n/k)^k.

Original entry on oeis.org

1, 3, 5, 10, 12, 26, 28, 52, 73, 115, 117, 295, 297, 439, 713, 1160, 1162, 2448, 2450, 4644, 6832, 8902, 8904, 23536, 25639, 33857, 53247, 84961, 84963, 192237, 192239, 318477, 493909, 625015, 695789, 1761668, 1761670, 2285996, 3872598, 6255230, 6255232, 13392362

Offset: 1

Seiichi Manyama, Jun 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A001923, A006218, A332469, A345098, A345100.

a[n_] := Sum[Floor[n/k]^k, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jun 10 2021 *)

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

my(N=66, x='x+O('x^N)); Vec(sum(j=1, N, (1-x^j)*sum(k=1, N, (k*x^k)^j))/(1-x))

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

a(n) ~ 3^((n - mod(n,3))/3 + 1)/2. - Vaclav Kotesovec, Jun 11 2021

A074309 Sum of next n terms of the form i^i.

Original entry on oeis.org

4, 283, 50037, 17650540, 10405067904, 9211817140115, 11424093748466841, 18896062057822100616, 40192544399240309019728, 106876212200059543898143707, 347377340594805599176614321101

Offset: 1

Zak Seidov, Sep 22 2002

nonn

Sum of next n terms of the form i^n is A074209. Sum of first n terms of the form i^n is A031971. Sum of first n terms of the form i^i is A001923.

			a(2) = 283 = 3^3 + 4^4, a(3) = 50037 = 4^4 + 5^5 + 6^6, a(4) = 17650540 = 5^5 + 6^6 + 7^7 + 8^8, a(5) = 10405067904 = 6^6 + 7^7 + 8^8 + 9^9 + 10^10.

Seiichi Manyama, Table of n, a(n) for n = 1..193

Cf. A001923, A031971, A074209.

Mathematica
```
Table[Sum[i^i, {i, n+1, 2n}], {n, 20}]
```

a(n) = sum(k=n+1, 2*n, k^k); \\ Seiichi Manyama, Dec 05 2021

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

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

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

cp's OEIS Frontend

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

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A188776 Numbers n such that Sum_{k=1..n} k^k == 1 (mod n).

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A342449 a(n) = Sum_{k=1..n} gcd(k,n)^k.

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A344430 a(n) = Sum_{k=1..n} mu(k) * k^k.

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A074209 a(n) = Sum_{i=n+1..2n} i^n.

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A122166 Numbers n such that 1 + Sum k^k (k=1..n) is prime.

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A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

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A345176 a(n) = Sum_{k=1..n} floor(n/k)^k.

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