A188775 -id:A188775 - cp's OEIS frontend

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

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

A341437 Numbers k such that k divides Sum_{j=0..k} j^(k-j).

Original entry on oeis.org

1, 2, 6, 7, 9, 42, 46, 431, 1806, 2506, 11318, 16965, 25426, 33146, 33361, 37053, 49365, 99221, 224506, 359703, 436994

Offset: 1

Seiichi Manyama, Feb 11 2021

Numbers k such that k divides A026898(k-1).

a(19) > 10^5.

Cf. A026898, A128981, A188775, A341436.

Do[If[Mod[Sum[PowerMod[k, n - k, n], {k, 0, n}], n] == 0, Print[n]], {n, 1, 3000}] (* Vaclav Kotesovec, Feb 12 2021 *)

isok(n) = sum(k=0, n, Mod(k, n)^(n-k))==0;

0^6 + 1^5 + 2^4 + 3^3 + 4^2 + 5^1 + 6^0 = 66 = 6 * 11. So 6 is a term.

a(19) from Vaclav Kotesovec, Feb 14 2021

a(20)-a(21) from Chai Wah Wu, Feb 15 2021

A341436 Numbers k such that k divides Sum_{j=1..k} j^(k+1-j).

Original entry on oeis.org

1, 5, 16, 208, 688, 784, 2864, 9555, 17776, 81239

Offset: 1

Seiichi Manyama, Feb 11 2021

Numbers k such that k divides A003101(k).

a(11) > 10^5.

			1^5 + 2^4 + 3^3 + 4^2 + 5^1 = 65 = 5 * 13. So 5 is a term.

Mathematics Stack Exchange, Is it true that if p != 5 is a prime number then 1^p + 2^(p-1) + ... + (p-1)^2 + p^1 != 0 (mod p)?.

Cf. A003101, A128981, A188775, A341437.

Do[If[Mod[Sum[PowerMod[k, n + 1 - k, n], {k, 1, n}], n] == 0, Print[n]], {n, 1, 3000}] (* Vaclav Kotesovec, Feb 12 2021 *)

isok(n) = sum(k=1, n, Mod(k, n)^(n+1-k))==0;

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.

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

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A341437 Numbers k such that k divides Sum_{j=0..k} j^(k-j).

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A341436 Numbers k such that k divides Sum_{j=1..k} j^(k+1-j).

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

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