A062727 -id:A062727 - cp's OEIS frontend

A366819 a(n) is the sum of the divisors of n^n-1.

4, 42, 432, 6048, 67584, 1704240, 38054016, 967814400, 16203253248, 513593801496, 15743437516800, 720045832568832, 19146847615988736, 835966563470742528, 31421980989189888768, 1602925310146310674200, 52064744760120508416000, 4286575920597346109768658

Offset: 2

Sean A. Irvine, Oct 24 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 2..138

Cf. A000203, A048861, A309941, A344870, A334167, A062727, A366820.

Array[DivisorSigma[1, #^# - 1] &, 18, 2] (* Michael De Vlieger, Oct 24 2023 *)

PARI
```
a(n) = sigma(n^n-1);
```

a(n) = A000203(A048861(n)).

A275123 Even numbers n such that sigma(n) divides sigma(n^n).

Original entry on oeis.org

4, 16, 64, 100, 196, 484, 676, 1024, 1156, 1296, 1444, 1936, 2116, 3364, 3844, 4096, 4900, 5476, 5776, 6400, 6724, 7396, 8836, 10816, 11236, 12100, 13456, 13924, 14884, 15376, 16900, 17956, 20164, 21316, 23716, 24964, 26896, 27556, 28900, 31684, 33124, 36100

Offset: 1

Altug Alkan, Jul 18 2016

nonn

A number n with prime factorization Product_i p_i^(e_i) is in the sequence iff Product_i ((p_i^(e_i*n+1)-1)/(p_i^(e_i+1)-1)) is an integer. - Robert Israel, Jul 19 2016

Does this sequence consist of the even numbers n such that A000005(n) divides A000005(n^n)? The answer is no according to the b-file since 50176 is missing (((2^(10*50176+1)-1)*(7^(2*50176+1)-1)) mod ((2^11-1)*(7^3-1)) = 372438 and (10*50176+1)*(2*50176+1) mod (11*3) = 0). Note that 50176 is the least number with this property.

			4 is a term because sigma(4^4) = 511 is divisible by sigma(4) = 7.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A000203, A062727.

filter:= proc(n) local F,t,b,r;
  F:= ifactors(n)[2];
  b:= mul(t[1]^(t[2]+1)-1, t=F);
  r:= 1;
  for t in F do r:= r * (t[1] &^ (t[2]*n+1)-1) mod b od;
  r = 0;
end proc:
select(filter, [seq(i,i=2..10^5,2)]); # Robert Israel, Jul 19 2016

Select[Range[2, 10^4, 2], Divisible[DivisorSigma[1, #^#], DivisorSigma[1, #]] &] (* Michael De Vlieger, Jul 19 2016 *)

/* Requires a large PARI stack to return even the first few terms */
is(n) = Mod(n, 2)==0 && Mod(sigma(n^n), sigma(n))==0 \\ Felix Fröhlich, Jul 19 2016

a(8)-a(22) from Michel Marcus, Jul 19 2016

More terms from Robert Israel, Jul 19 2016

A275391 Least k such that n divides sigma(k^k) (k > 0).

Original entry on oeis.org

1, 3, 5, 3, 3, 5, 2, 3, 5, 3, 19, 11, 11, 5, 15, 7, 15, 5, 11, 3, 5, 19, 10, 11, 7, 11, 17, 11, 13, 15, 5, 7, 29, 15, 23, 11, 11, 11, 11, 3, 15, 5, 35, 19, 23, 21, 22, 15, 13, 7, 15, 11, 23, 17, 19, 11, 11, 13, 28, 15, 11, 5, 5, 15, 15, 29, 21, 15, 65, 23, 34, 11, 4, 11, 29, 11, 39, 11, 23, 7, 17

Offset: 1

Altug Alkan, Aug 07 2016

nonn

From Robert Israel, Aug 09 2016: (Start)
a(n) <= A038700(n) if n >= 4, since sigma(k^k) == 0 (mod n) if k is an odd prime == -1 (mod n).
If n is prime and n-2 is squarefree, then a(n) <= n-2 since sigma((n-2)^(n-2)) == 0 (mod n).
Conjecture: a(n) <= n-2 for all n > 15, but a(n) = n-2 for infinitely many n. (End)

			a(11) = 19 because sigma(19^19) is divisible by 11.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A038700, A062727, A275800.

N:= 200: # to get a(1)..a(N)
S:= {$1..N}:
for k from 1 while S <> {} do
  v:= numtheory:-sigma(k^k);
  F:= select(t -> v mod t = 0, S);
  for n in F do A[n]:= k od:
  S:= S minus F;
od:
seq(A[n],n=1..N); # Robert Israel, Aug 09 2016

a(n) = {my(k=1); while(sigma(k^k) % n != 0, k++); k; }

A366820 a(n) is the sum of the divisors of n^n + 1.

Original entry on oeis.org

3, 3, 6, 56, 258, 6264, 52136, 1559520, 17041416, 706911048, 10102223208, 706019328000, 9101898907920, 519285252355776, 11672709747324912, 880565163670372352, 18446811354131136516, 1792353900753729655758, 54357680125881245248800, 4154723599066412190910560

Offset: 0

Sean A. Irvine, Oct 24 2023

nonn

Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind

Cf. A000203, A007571, A014566, A055385, A062727, A344859, A366819.

{3}~Join~Array[DivisorSigma[1, #^# + 1] &, 19] (* Michael De Vlieger, Oct 24 2023 *)

PARI
```
a(n) = sigma(n^n+1);
```

a(n) = A000203(A014566(n)).

A275800 n such that A275391(n) = n-2.

Original entry on oeis.org

5, 13, 17, 139, 173, 179, 467, 907, 1553, 1619, 1867, 2099, 2819, 2957, 3203, 3779, 3947, 4139, 4157, 4283, 4547, 4723, 5483, 6653, 6899, 7013, 7187, 7523, 7643, 8147, 8387, 8563, 8573, 8753, 9533, 9587, 10589, 10853, 10883, 10979, 11003, 12107, 12227, 13037, 13229, 13829, 14243, 14549, 14699, 14867, 15299, 16217, 16547, 16649, 17387, 18443, 18587, 19259

Offset: 1

Robert Israel, Aug 09 2016

nonn

n such that n-2 is the least k such that n divides A062727(k) = sigma(k^k).

Are all terms prime?

			17 is in the sequence because 17 divides sigma(15^15) = 821051025385244160 but does not divide sigma(k^k) for any k < 15.

Cf. A000203, A062727, A275391.

N:= 20000:
S:= {$1..N}: # to get terms <= N
for kk from 1 while S <> {} do
   v:= numtheory:-sigma(kk^kk);
   F:= select(t -> v mod t = 0, S);
   for nn in F do
     B[nn]:= kk
   od;
   S:= S minus F;
od:
select(t -> B[t]=t-2, [$1..N]);

A309377 a(n) is the product of the divisors of n^n (A000312).

Original entry on oeis.org

1, 1, 8, 729, 68719476736, 30517578125, 2444746349972956194083608044935243159422957210683702349648543934214737968217920868940091707112078529114392164827136, 459986536544739960976801, 2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397376

Offset: 0

Hauke Löffler, Jul 26 2019

nonn

Subset of A007955.

			a(0) = 1 because 0^0 = 1, whose only divisor is 1, so the product of divisors is 1.
a(1) = 1 because 1^1 = 1, so the product of divisors is 1.
a(3) = 729 because 3^3 = 27, whose divisors are (1, 3, 9, 27), and their product is 729.

Cf. A007955, A062727.

[&*Divisors(n^n): n in [0..8]]; // Marius A. Burtea, Jul 26 2019

from math import isqrt, prod
from sympy import factorint
def A309377(n): return (isqrt(n**n) if (c:=prod(n*e+1 for e in factorint(n).values())) & 1 else 1)*n**(n*(c//2)) # Chai Wah Wu, Jun 25 2022

[ product((1*i^i).divisors()) for i in range(10) ]

cp's OEIS Frontend

A366819 a(n) is the sum of the divisors of n^n-1.

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A275123 Even numbers n such that sigma(n) divides sigma(n^n).

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A275391 Least k such that n divides sigma(k^k) (k > 0).

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A366820 a(n) is the sum of the divisors of n^n + 1.

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Mathematica

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A275800 n such that A275391(n) = n-2.

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Maple

A309377 a(n) is the product of the divisors of n^n (A000312).

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