A188776 -id:A188776 - 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

A343983 Numbers k such that Sum_{j|k} j^j == 1 (mod k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 72, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Seiichi Manyama, May 06 2021

nonn

This sequence is different from A074583.

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

Cf. A062796, A188776.

q[n_] := Divisible[DivisorSum[n, #^# &] - 1, n]; Select[Range[260], q] (* Amiram Eldar, May 06 2021 *)

isok(n) = sumdiv(n, d, Mod(d, n)^d)==1;

from itertools import count, islice
from sympy import divisors
def A343983_gen(): # generator of terms
    yield 1
    for k in count(1):
        if sum(pow(j,j,k) for j in divisors(k,generator=True)) % k == 1:
            yield k
A343983_list = list(islice(A343983_gen(),30)) # Chai Wah Wu, Jun 19 2022

A343930 Numbers k such that Sum_{j=1..k} (-j)^j == 1 (mod k).

Original entry on oeis.org

1, 2, 30, 33, 37, 83, 149, 262, 4030, 31969, 140225, 182730, 724754, 2337094, 3985753, 4195221, 4541725

Offset: 1

Seiichi Manyama, May 04 2021

Cf. A188776, A341437, A343931, A343933.

q[n_] := n == 1 || Mod[Sum[PowerMod[-k, k, n], {k, 1, n}], n] == 1; Select[Range[5000], q] (* Amiram Eldar, May 04 2021 *)

PARI
```
isok(n) = sum(k=1, n, Mod(-k, n)^k)==1;
```

a(11)-a(13) from Chai Wah Wu, May 04 2021
a(14) from Martin Ehrenstein, May 05 2021
a(15)-a(17) from Martin Ehrenstein, May 08 2021

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

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

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A343983 Numbers k such that Sum_{j|k} j^j == 1 (mod k).

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A343930 Numbers k such that Sum_{j=1..k} (-j)^j == 1 (mod k).

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

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