A014566 -id:A014566 - cp's OEIS frontend

A342490 a(n) = Sum_{d|n} phi(d)^(n-1).

1, 2, 5, 10, 257, 66, 46657, 16514, 1679873, 524290, 10000000001, 4200450, 8916100448257, 26121388034, 4398314962945, 35185445863426, 18446744073709551617, 33853319151618, 39346408075296537575425, 144115737832194050, 3833763648605916233729, 2000000000000000000002

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A000010, A014566, A342471, A342487.

a[n_] := DivisorSum[n, EulerPhi[#]^(n-1) &]; Array[a, 20] (* Amiram Eldar, Mar 14 2021 *)

a(n) = sumdiv(n, d, eulerphi(d)^(n-1));

a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n-2));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^(k-1)*x^k/(1-(eulerphi(k)*x)^k)))

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(n-2).
G.f.: Sum_{k>=1} phi(k)^(k-1) * x^k/(1 - (phi(k) * x)^k).
If p is prime, a(p) = 1 + (p-1)^(p-1) = A014566(p-1).

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)).

A064772 Sum of primes dividing n^n+1 (with repetition).

Original entry on oeis.org

2, 5, 11, 257, 526, 147, 1030, 1027, 530793, 31603, 58685, 2228292, 113060, 180326, 163123, 67280421584898, 45957792327018709129, 33414185, 870543318650, 4406613081350403, 22864393425065, 82812579069940, 1576297793, 27266016518

Offset: 1

Jason Earls, Oct 19 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..61

Cf. A014566, A001414.

Table[Total[Times@@@FactorInteger[n^n+1]],{n,25}] (* Harvey P. Dale, Sep 17 2011 *)

sopfr(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]*f[i, 2]); return(s) } { for (n=1, 61, write("b064772.txt", n, " ", sopfr(n^n + 1)) ) } \\ Harry J. Smith, Sep 24 2009

A073943 a(n) = smallest m such that n-th prime divides m^m + 1.

Original entry on oeis.org

1, 5, 2, 3, 21, 6, 24, 27, 11, 14, 15, 6, 25, 63, 23, 26, 117, 30, 99, 35, 9, 39, 165, 11, 6, 10, 51, 213, 54, 7, 63, 261, 174, 23, 74, 33, 78, 27, 83, 86, 357, 30, 95, 12, 14, 11, 15, 111, 453, 18, 12, 119, 90, 501, 4, 131, 82, 135, 117, 60, 45, 138, 51, 95, 54, 114, 75

Offset: 1

Jason Earls, Nov 13 2002

			6th prime is 13 and 13 first divides 6^6 + 1 = 46657, so a(6) = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A014566.

smp[n_]:=Module[{m=1},While[PowerMod[m,m,n]!=n-1,m++];m]; smp/@Prime[ Range[ 70]] (* Harvey P. Dale, Jul 21 2018 *)

A085602 Numbers of the form (2n+1)^(2n+1) + 1.

Original entry on oeis.org

2, 28, 3126, 823544, 387420490, 285311670612, 302875106592254, 437893890380859376, 827240261886336764178, 1978419655660313589123980, 5842587018385982521381124422, 20880467999847912034355032910568, 88817841970012523233890533447265626

Offset: 1

Cino Hilliard, Jul 07 2003

Also even Sierpinski numbers of the first kind.

No term is a square. Moreover, x^x + 1 != k^x, for if it were, we would have a counterexample to Fermat's Last Theorem.

Harvey P. Dale, Table of n, a(n) for n = 1..193

Bisection of A014566 (odd part).

#^#+1&/@Range[1,21,2] (* Harvey P. Dale, Dec 08 2012 *)

PARI
```
forstep(x=1,20,2,print1(x^x+1" "))
```

a(n) = (2*n-1)^(2*n-1)+1. - Alois P. Heinz, Feb 27 2020

A131839 Additive persistence of Sierpinski numbers of first kind.

Original entry on oeis.org

0, 0, 2, 2, 2, 3, 2, 3, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 3, 3, 4, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 2, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 1, 3, 4, 3, 3, 4

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

			Sierpinski number 257 --> 2+5+7 = 14 --> 1+4 = 5 thus persistence is 2.
The sixteenth Sierpinski number is 16^16 + 1 = 18446744073709551617 --> 1+8+4+4+6+7+4+4+0+7+3+7+0+9+5+5+1+6+1+7 = 89 --> 8+9 = 17 --> 1+7 = 8, thus a(16) = 3 because in three steps we obtain a number < 10. - _Antti Karttunen_, Dec 15 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 2048 terms from Antti Karttunen)

Cf. A014566, A131836.

f:= proc(n) local t, count;
  t:= n^n+1;
  count:= 0;
  while t > 9 do
    count:= count+1;
    t:= convert(convert(t,base,10),`+`);
  od;
  count
end proc:
map(f, [$1..100]); # Robert Israel, Dec 18 2017

f[n_] := Length@ NestWhileList[Plus @@ IntegerDigits@# &, n^n + 1, UnsameQ@## &,     All] - 2; Array[f, 105] (* Robert G. Wilson v, Dec 18 2017 *)

allocatemem(2^30);
A007953(n) = { my(s); while(n, s+=n%10; n\=10); s; };
A031286(n) = { my(s); while(n>9, s++; n=A007953(n)); s; }; \\ This function after Charles R Greathouse IV, Sep 13 2012
A014566(n) = (1+(n^n));
A131839(n) = A031286(A014566(n)); \\ Antti Karttunen, Dec 15 2017

a(n) = A031286(A014566(n)). - Antti Karttunen, Dec 15 2017

Erroneous terms (first at n=16) corrected by Antti Karttunen, Dec 15 2017

A225945 Numbers k such that prime(k) divides k^k + 1.

Original entry on oeis.org

1, 6, 60, 136, 124796, 3919272, 18363918, 153037808, 965108649, 3140421892, 5961162423, 20437804784

Offset: 1

Alex Ratushnyak, May 21 2013

a(12) > 2*10^10. - Giovanni Resta, May 23 2013

Cf. A000040, A000312, A014566, A225944.

Select[Range[10^6], (p = Prime[#]; PowerMod[#, #, p] == p - 1) &] (* Giovanni Resta, May 23 2013 *)

from sympy import nextprime, prime
from itertools import count, islice
def agen(startn=1): # generator of terms
    pn = prime(startn)
    for n in count(startn):
        if pow(n, n, pn) == pn - 1:
            yield n
        pn = nextprime(pn)
print(list(islice(agen(), 5))) # Michael S. Branicky, May 25 2023

a(6)-a(11) from Giovanni Resta, May 23 2013

a(12) from Michael S. Branicky, May 25 2023

A271186 Odd integers k such that k^k + 1 is the sum of 2 nonzero squares.

Original entry on oeis.org

1, 9, 17, 25, 49, 73, 81, 89, 97, 121

Offset: 1

Altug Alkan, Apr 01 2016

			9 is a term because 9^9 + 1 = 1457^2 + 19629^2.

Cf. A000404, A014566.

Select[Range[1, 25, 2], Length[PowersRepresentations[#^# + 1, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 01 2016 *)

a014566(n) = n^n+1;
isA000404(n) = { for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
for(n=1, 1e2, if(isA000404(a014566(n)) && n%2 == 1, print1(n, ", ")));

a(6)-a(10) from Jinyuan Wang, Aug 14 2022

A271267 Even numbers k such that k + 2 divides k^k + 2.

Original entry on oeis.org

4, 16, 196, 2836, 5956, 25936, 65536, 540736, 598816, 797476, 1151536, 3704416, 8095984, 11272276, 13362420, 21235696, 29640832, 31084096, 42913396, 49960912, 55137316, 70254724, 70836676, 81158416, 94618996, 111849956, 129275056, 150026176, 168267856, 169242676, 189796420, 192226516, 198464176, 208232116, 244553296, 246605776, 300018016, 318143296

Offset: 1

Altug Alkan, Apr 03 2016

nonn

In other words, even numbers k such that k + 2 divides A014566(k) + 1.
Even terms of A213382.
4, 16, 65536 are the numbers of the form 2^(2^(2^k)), for k >= 0. Are there other members of this sequence with the form of 2^(2^(2^k))?
2^(2^(2^3)) and 2^(2^(2^4)) are terms. - Michael S. Branicky, Apr 16 2021

			4 is a term because 4 + 2 = 6 divides 4^4 + 2 = 258.

Michael S. Branicky, Table of n, a(n) for n = 1..150

Cf. A014566, A081765, A213382.

Select[Range[2, 10^4, 2], Divisible[#^# + 2, # + 2] &] (* Michael De Vlieger, Apr 03 2016 *)

lista(nn) = forstep(n=2, nn, 2, if( Mod(n, n+2)^n == -2 , print1(n, ", "))); \\ Joerg Arndt, Apr 03 2016

def afind(limit):
  k = 2
  while k < limit:
    if (pow(k, k, k+2) + 2)%(k+2) == 0: print(k, end=", ")
    k += 2
afind(10**7) # Michael S. Branicky, Apr 16 2021

A009661 Smallest number m such that m^m+1 is divisible by n.

Original entry on oeis.org

0, 0, 5, 3, 2, 5, 3, 7, 17, 9, 21, 11, 6, 3, 29, 15, 24, 17, 27, 19, 41, 21, 11, 23, 18, 25, 53, 3, 14, 29, 15, 31, 35, 33, 69, 35, 6, 27, 77, 39, 25, 41, 63, 35, 89, 11, 23, 47, 97, 49, 101, 51, 26, 53, 109, 55, 113, 35, 117, 59, 30, 15, 125, 63, 18, 35, 99, 67, 11, 69, 35, 71, 9, 27

Offset: 1

David W. Wilson

nonn

If n is odd, then a(n) <= 2*n - 1. If n is even, then a(n) <= n - 1. - Thomas Ordowski, Dec 03 2023

Cf. A014566, A133090, A268466.

a(n) = my(m=0); while ((1+Mod(m, n)^m) != 0, m++); m; \\ Michel Marcus, Dec 03 2023

cp's OEIS Frontend

A342490 a(n) = Sum_{d|n} phi(d)^(n-1).

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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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A064772 Sum of primes dividing n^n+1 (with repetition).

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A073943 a(n) = smallest m such that n-th prime divides m^m + 1.

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Mathematica

A085602 Numbers of the form (2n+1)^(2n+1) + 1.

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Mathematica

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A131839 Additive persistence of Sierpinski numbers of first kind.

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Maple

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A225945 Numbers k such that prime(k) divides k^k + 1.

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A271186 Odd integers k such that k^k + 1 is the sum of 2 nonzero squares.

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