A014566 -id:A014566 - cp's OEIS frontend

A122000 a(n) = ((2^n - 1)^(2^n - 1) + 1) / 2^n.

1, 7, 102943, 27368368148803711, 533411691585101123706582594658103586126397951, 3566766192921360077810945505268211287512797261288920841093043641769808083046939618603793791988232043305924036607

Offset: 1

Alexander Adamchuk, Sep 11 2006

nonn

A014566(n) = n^n + 1 is Sierpinski Number of the First Kind. A014566(2^n - 1) is divisible by 2^n. a(n) is a subset of A081216(n) = (n^n-(-1)^n)/(n+1).

2^p - 1 divides a(p-1) for prime p>2. Corresponding quotients are a(p-1) / (2^p - 1) = {1, 882850585445281, 28084773172609134470952326813135521948919663474715912134590894817085103016117634792155856629828598766188378241, ...}, where p = prime(n) for n>1. - Alexander Adamchuk, Jan 22 2007

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

Cf. A014566, A081216, A056009.

Mathematica
```
Table[((2^n-1)^(2^n-1)+1)/2^n,{n,1,7}]
```

a(n) = A014566(2^n - 1) / 2^n.
a(n) = A081216(2^n - 1).
a(n) = A056009(2^n - 1).

A134883 Decimal expansion of Sum_{n>=1} 1/(n^n+1).

Original entry on oeis.org

7, 3, 9, 9, 4, 7, 9, 4, 3, 4, 9, 5, 4, 6, 5, 5, 1, 2, 2, 5, 6, 0, 2, 5, 5, 3, 0, 7, 3, 4, 9, 9, 4, 7, 8, 2, 0, 5, 6, 1, 1, 0, 6, 6, 5, 7, 4, 2, 2, 4, 3, 9, 6, 2, 8, 7, 4, 5, 4, 5, 6, 5, 1, 9, 9, 9, 8, 0, 4, 3, 0, 8, 5, 4, 0, 8, 4, 8, 8, 1, 0, 2, 8, 9, 7, 3, 9, 5, 3, 1, 1, 2, 0, 7, 1, 2, 1, 5, 6, 8, 2, 0, 5, 9

Offset: 0

Artur Jasinski, Nov 15 2007

Constant formed from sum of reversed Sierpinski numbers of first kind A014566.

			0.7399479434954655122560255307349947820561106657422439628745456519998...

Cf. A014566, A073009, A100085, A134877, A134878, A134879, A134880, A134881, A134882.

RealDigits[N[Sum[1/(n^n + 1), {n, 1, 150}], 100]][[1]] (* first zero removed *)

A184967 Numbers k such that k^k+1 is squarefree.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 16, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 41, 42, 44, 45, 46, 48, 50, 52, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 72, 73, 74, 76, 77, 78, 80, 81, 84, 85, 86, 88, 90, 92, 93, 94, 96, 98, 100, 101, 102, 104, 105, 106, 108, 109, 110, 112, 113, 114, 116, 120, 121, 122

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 19 2011

nonn

5^5+1 = 3126 = 2*3*521.
6^6+1 = 46657 = 13*37*97.
8^8 = 16777217 = 97*257*673.

Cf. A005117, A014566, A184966.

[ n: n in [1..60] | IsSquarefree(n^n+1) ]; // Bruno Berselli, Mar 24 2011

Mathematica
```
Select[Range@46,SquareFreeQ[#^#+1]&]
```

isok(k) = issquarefree(k^k+1); \\ Michel Marcus, Feb 22 2021

Extended by Bruno Berselli, Mar 24 2011

a(39)-a(78) from Amiram Eldar, Feb 22 2021

A238981 Sum of n-th powers of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

Original entry on oeis.org

1, 5, 28, 257, 3126, 47450, 823544, 16777217, 387420490, 10009766650, 285311670612, 8916117756914, 302875106592254, 11112685048647250, 437893920912786408, 18446744073709551617, 827240261886336764178, 39346558169931834836690, 1978419655660313589123980

Offset: 1

José María Grau Ribas, Mar 07 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..387

Cf. A034448, A238982, A238983.

a[n_, k_] := Sum[If[GCD[i, n] == i && GCD[i, n/i] == 1, i^k, 0], {i, n}]; Table[a[n, n], {n, 1, 24}]
a[1] = 1; a[n_] := Times @@ (1 + First[#]^(n * Last[#]) &/@ FactorInteger[n]);  Array[a, 19] (* Amiram Eldar, Aug 10 2019 *)

a(n) = sumdiv(n, d, d^n*(gcd(d, n/d) == 1)); \\ Michel Marcus, Mar 19 2014

For prime p, a(p) = p^p + 1; A125137 is a subsequence. - Michel Marcus, Nov 20 2015
a(n) = n^n+1 (A014566) if n is a prime power (A246655). - Michel Marcus, Nov 21 2015
a(n) = Sum_{d|n, gcd(d,n/d)=1} d^n. - Wesley Ivan Hurt, Apr 28 2023

A342489 a(n) = Sum_{d|n} phi(d)^(d-1).

Original entry on oeis.org

1, 2, 5, 10, 257, 38, 46657, 16394, 1679621, 262402, 10000000001, 4194350, 8916100448257, 13060740674, 4398046511365, 35184372105226, 18446744073709551617, 16926661124390, 39346408075296537575425, 144115188076118282, 3833759992447475168837

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

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

Cf. A000010, A014566, A164941, A342473, A342485.

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

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

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

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

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

A342707 T(n, k) is the result of replacing 2's by k's in the hereditary base-2 expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 2, 1, 3, 3, 1, 0, 1, 2, 4, 4, 4, 1, 0, 2, 2, 5, 27, 5, 5, 1, 0, 0, 3, 6, 28, 256, 6, 6, 1, 0, 1, 1, 7, 30, 257, 3125, 7, 7, 1, 0, 0, 2, 8, 31, 260, 3126, 46656, 8, 8, 1, 0, 1, 2, 9, 81, 261, 3130, 46657, 823543, 9, 9, 1, 0

Offset: 0

Rémy Sigrist, Jun 04 2021

			Array T(n, k) begins:
  n\k|  0  1   2   3     4      5       6        7          8           9
  ---+-------------------------------------------------------------------
    0|  0  0   0   0     0      0       0        0          0           0
    1|  1  1   1   1     1      1       1        1          1           1
    2|  0  1   2   3     4      5       6        7          8           9
    3|  1  2   3   4     5      6       7        8          9          10
    4|  1  1   4  27   256   3125   46656   823543   16777216   387420489
    5|  2  2   5  28   257   3126   46657   823544   16777217   387420490
    6|  1  2   6  30   260   3130   46662   823550   16777224   387420498
    7|  2  3   7  31   261   3131   46663   823551   16777225   387420499
    8|  0  1   8  81  1024  15625  279936  5764801  134217728  3486784401
    9|  1  2   9  82  1025  15626  279937  5764802  134217729  3486784402
   10|  0  2  10  84  1028  15630  279942  5764808  134217736  3486784410

Wikipedia, Hereditary base-n notation

See A341907 for a similar sequence.

Cf. A000120, A000312, A002488, A014566, A066068, A066279, A222112, A343255, A345021.

T(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n,2); v+=k^T(e,k)); v }

T(n, n) = A343255(n).
T(n, 0) = A345021(n).
T(n, 1) = A000120(n).
T(n, 2) = n.
T(n, 3) = A222112(n-1).
T(0, k) = 0.
T(1, k) = 1.
T(2, k) = k.
T(3, k) = k + 1.
T(4, k) = k^k = A000312(k).
T(5, k) = k^k + 1 = A014566(k).
T(6, k) = k^k + k = A066068(k).
T(7, k) = k^k + k + 1 = A066279(k).
T(16, k) = k^k^k = A002488(k).
T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).

A366822 a(n) is phi(n^n + 1) where phi is the Euler totient function.

Original entry on oeis.org

1, 1, 4, 12, 256, 1040, 41472, 407680, 16515072, 152845056, 9897840000, 89493288192, 8732596764672, 129785922489600, 10576701872701440, 210729768933600000, 18446676793287966720, 275746753962112254720, 28084363369373740400640, 791359800910482004224000

Offset: 0

Sean A. Irvine, Oct 24 2023

nonn

Robert G. Wilson v, Table of n, a(n) for n = 0..72
Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind

Cf. A000010, A014566, A064447, A366821.

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

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

a(n) = A000010(A014566(n)).

A117812 a(n) = n^(2*n) - 1.

Original entry on oeis.org

0, 0, 15, 728, 65535, 9765624, 2176782335, 678223072848, 281474976710655, 150094635296999120, 99999999999999999999, 81402749386839761113320, 79496847203390844133441535, 91733330193268616658399616008, 123476695691247935826229781856255

Offset: 0

Reinhard Zumkeller, Oct 17 2006

nonn

a(n) = A048861(n)*A014566(n) = A062206(n) - 1.

Cf. A000312, A014566, A048861, A062206.

[n^(2*n) - 1: n in [0..40]]; // Vincenzo Librandi, Dec 26 2010

Join[{0},Table[(n^2)^n-1,{n,17}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)

A131836 Multiplicative persistence of the Sierpinski numbers of the first kind (n^n + 1).

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

Question: Are there any terms larger than 1 after a(22) = 2? In other words, do all terms of A014566 contain zero somewhere in their decimal representation after A014566(22) = 341427877364219557396646723585? - Antti Karttunen, Oct 08 2017

			For n=4 we have A014566(4) = Sierpinski number 257 --> 2*5*7 = 70 --> 7*0 = 0 thus persistence = 2, and a(4) = 2. - Edited by _Antti Karttunen_, Oct 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A014566, A031346.

P:=proc(n) local i,k,w,ok,cont; for i from 1 by 1 to n do w:=1; k:=i^i+1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(100);

Table[-1 + Length@ NestWhileList[Times @@ IntegerDigits@ # &, If[n == 0, 2, n^n + 1], # > 9 &], {n, 105}] (* Michael De Vlieger, Oct 08 2017 *)

;; The whole program follows:
(define (A131836 n) (A031346 (A014566 n)))
(define (A014566 n) (+ 1 (expt n n)))
(define (A031346 n) (let loop ((n n) (k 0)) (if (< n 10) k (loop (A007954 n) (+ 1 k)))))
(define (A007954 n) (if (zero? n) n (let loop ((n n) (m 1)) (if (zero? n) m (let ((d (modulo n 10))) (loop (/ (- n d) 10) (* d m)))))))
;; Antti Karttunen, Oct 08 2017

a(n) = A031346(A014566(n)). - Michel Marcus, Oct 08 2017

A249784 Number of divisors of n^(n^n).

Original entry on oeis.org

1, 5, 28, 513, 3126, 2176875649, 823544, 50331649, 774840979, 100000000020000000001, 285311670612, 158993694406808436568227841, 302875106592254, 123476695691247958050243432972289, 191751059232884087544279144287109376, 73786976294838206465

Offset: 1

Jon E. Schoenfield, Nov 05 2014

nonn

An infinite number of squares are terms of this sequence.
Proof: for any n of the form (p*q)^k (with p and q distinct primes), a(n) = (k * n^n + 1)^2.
It seems likely that the only nontrivial palindromes in this sequence comprise a subset of these squares and occur at n = 10^(10^M) for M>=0; at such values of n, a(n) = (10^(10^(10^M + M) + M) + 1)^2 = A033934(10^(10^M + M) + M). The actual decimal expansion of each of these numbers is of the form 1000...0002000...0001, where the total number of zero digits on each side of the 2 is 10^(10^M + M) + M - 1.

			12 = 2^2 * 3^1 (two distinct prime factors, with multiplicities e_1=2 and e_2=1), so a(12) = (2*k+1)*(1*k+1) = 2*k^2 + 3*k + 1 where k = 12^12, so a(12) = 158993694406808436568227841.

Jon E. Schoenfield, Table of n, a(n) for n = 1..155

Cf. A000005, A000312, A002488, A014566, A033934, A225739.

// program to generate b-file
for n in [1..155] do
   k:=n^n;
   F:=Factorization(n);
   prod:=1;
   for j in [1..#F] do
      prod*:=F[j,2]*k + 1;
   end for;
   n, prod;
end for;

a(n)=my(v=factor(n)[,2]*n^n); prod(i=1,#v,v[i]+1) \\ Charles R Greathouse IV, Jul 21 2015

def A249784(n):
   n_exp_n = n^n
   return mul(exp[1]*n_exp_n + 1 for exp in factor(n))
[A249784(n) for n in (1..16)] # Peter Luschny, Nov 08 2014

a(n) = A000005(A002488(n)).
a(n) = Product_{j=1..m} (e_j * n^n + 1)
   where m = number of distinct prime factors of n
   and e_j = multiplicity of the j-th prime factor.
If n is a prime p, then m=1 and e_1=1, so
   a(p) = p^p + 1 = A000312(p) + 1 = A014566(p).
If n=10^L, then m=2 and e_1=e_2=L, so
   a(10^L) = (L * 10^(L * 10^L) + 1)^2.

cp's OEIS Frontend

A122000 a(n) = ((2^n - 1)^(2^n - 1) + 1) / 2^n.

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Mathematica

Formula

A134883 Decimal expansion of Sum_{n>=1} 1/(n^n+1).

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Mathematica

A184967 Numbers k such that k^k+1 is squarefree.

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Magma

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PARI

Extensions

A238981 Sum of n-th powers of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

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Mathematica

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Formula

A342489 a(n) = Sum_{d|n} phi(d)^(d-1).

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Mathematica

PARI

PARI

PARI

Formula

A342707 T(n, k) is the result of replacing 2's by k's in the hereditary base-2 expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

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PARI

Formula

A366822 a(n) is phi(n^n + 1) where phi is the Euler totient function.

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Mathematica

PARI

Formula

A117812 a(n) = n^(2*n) - 1.

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