A048861 -id:A048861 - cp's OEIS frontend

A116895 Least prime factor of n^n-1.

3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 13, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2

Offset: 2

Giovanni Resta, Mar 02 2006

nonn

If n is odd then a(n)=2; also, if n is even and not divisible by 3 then a(n)=3. - Zak Seidov, Mar 03 2006

			6^6-1=5*7*31*43, so a(6)=5.

Antti Karttunen, Table of n, a(n) for n = 2..16384

Cf. A006486, A007571, A048861.

Table[FactorInteger[GCD[n^n-1, 200! ]][[1,1]], {n, 2, 130}]

A116895(n) = { my(k=(n^n)-1); forprime(p=2, ,if(!(k%p),return(p))); }; \\ Antti Karttunen, Dec 19 2018

A184966 Numbers k such that k^k - 1 is squarefree.

Original entry on oeis.org

2, 3, 4, 6, 7, 11, 12, 14, 15, 16, 20, 22, 23, 27, 31, 34, 35, 36, 38, 39, 42, 43, 47, 52, 56, 58, 59, 60, 63, 66, 67, 70, 71, 72, 75, 78, 79, 83, 84, 86, 87, 88, 90, 92, 94, 95, 96, 102, 103, 104, 106, 107, 108, 110, 111, 112, 114, 115, 119, 123, 128, 131, 135, 138

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 19 2011

nonn

3^3 - 1 = 26 = 2 * 13.
4^4 - 1 = 255 = 3 * 5 * 17.
6^6 - 1 = 46655 = 5 * 7 * 31 * 43.

Cf. A005117, A048861, A184967.

Select[Range@43, SquareFreeQ[#^# - 1] &]

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

a(23)-a(64) from Amiram Eldar, Feb 22 2021

A366821 a(n) is phi(n^n-1) where phi is the Euler totient function.

Original entry on oeis.org

2, 12, 128, 1400, 30240, 264992, 6635520, 141087744, 5890320000, 114117380608, 4662793175040, 99053063903040, 5470524984113280, 167080949856000000, 9208981628670443520, 413582117375670921216, 29531731481729468006400, 659473218553437863041320

Offset: 2

Sean A. Irvine, Oct 24 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 2..138 (terms 2..102 from Robert G. Wilson v)

Cf. A000010, A048861, A064447, A309941, A334167, A344870, A366819, A366822.

a:= n-> numtheory[phi](n^n-1):
seq(a(n), n=2..20);  # Alois P. Heinz, Oct 26 2023

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

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

a(n) = A000010(A048861(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*)

A125556 Smallest prime p = n*m + 1 that divides m^m - 1 for some m > 1.

Original entry on oeis.org

3, 11, 61, 13, 31, 241, 43, 257, 271, 251, 1013, 673, 157, 71, 541, 241, 307, 883, 4409, 401, 1471, 463, 5843, 577, 601, 2081, 271, 757, 1973, 241, 2729, 2657, 4951, 1123, 21701, 937, 32191, 14479, 3433, 641, 1723, 2521, 5849, 6689, 1531, 6257, 18143

Offset: 1

Alexander Adamchuk, Jan 22 2007

nonn

Corresponding numbers m > 1 such that a(n) divides m^m - 1 are m = (a(n) - 1)/n = {2,5,20,3,6,40,6,32,30,25,92,56,12,5,36,15,18,...}.

Joseph L. Wetherell, Table of n, a(n) for n = 1..10000

Cf. A048861 (n^n - 1).

a(n) = forprime(p=3, , if (!((p-1) % n) && (m = (p-1)/n) && (m > 1) && !((m^m-1) % p), return (p));); \\ Michel Marcus, Nov 18 2017

Corrected a(47) and extended by Alexander Adamchuk, May 09 2010

A225944 Numbers k such that prime(k) divides k^k - 1.

Original entry on oeis.org

1, 2, 5, 124, 181, 696, 261800, 3834909, 18836480, 51432542, 69709961, 332054520, 3140421767

Offset: 1

Alex Ratushnyak, May 21 2013

a(14) > 10^12. - Giovanni Resta, May 11 2020

Cf. A000040, A000312, A048861, A225945.

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

primes = []
n = 0
def addPrime(k):
  global n
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
  n += 1
  if (n**n-1) % k == 0: print(n, end=", ")
addPrime(2)
addPrime(3)
for i in range(5, 10000000, 6):
  addPrime(i)
  addPrime(i+2)

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) == 1:
            yield n
        pn = nextprime(pn)
print(list(islice(agen(), 7))) # Michael S. Branicky, May 25 2023

a(8)-a(13) from Giovanni Resta, May 23 2013

A342930 Least positive number k such that n^n divides k*(k+1)/2.

Original entry on oeis.org

1, 7, 26, 511, 3124, 16767, 823542, 33554431, 387420488, 1787109375, 285311670610, 6737830608896, 302875106592252, 10190301669556224, 12913848876953124, 36893488147419103231, 827240261886336764176, 22831345258932427292672, 1978419655660313589123978, 35357007743740081787109375

Offset: 1

Seiichi Manyama, Mar 29 2021

			  n |     a(n) |         T(a(n)) = n^n * A342931(n).
----+----------+------------------------------------
  1 |        1 |               1 = 1^1 * 1.
  2 |        7 |              28 = 2^2 * 7.
  3 |       26 |             351 = 3^3 * 13.
  4 |      511 |          130816 = 4^4 * 511.
  5 |     3124 |         4881250 = 5^5 * 1562.
  6 |    16767 |       140574528 = 6^6 * 3013.
  7 |   823542 |    339111124653 = 7^7 * 411771.
  8 | 33554431 | 562949936644096 = 8^8 * 33554431.

David A. Corneth, Table of n, a(n) for n = 1..386 (terms < 10^1000)

Cf. A000217, A000312, A048861, A232097, A342931.

a(n) = my(k=1, m=n^n); while(k*(k+1)/2%m!=0, k++); k;

a(n) = { my(p = 2*n^n, f = factor(p), res = oo); for(i = 2^(#f~-1), 2^#f~-1, b = binary(i); pr = prod(j = 1, #f~, f[j,1]^(b[j]*f[j, 2])); ipr = p/pr; for(j = -1, 0, c = lift(chinese(Mod(-1-j, ipr), Mod(j, pr))); if(c > 0, res = min(res, c)))); res } \\ David A. Corneth, Mar 29 2021

a(p) = p^p - 1 for odd prime p. - David A. Corneth, Mar 29 2021

More terms from David A. Corneth, Mar 29 2021

A077098 Length of period of the continued fraction expansion of sqrt(-1+n^n).

Original entry on oeis.org

2, 1, 2, 10, 2, 26, 2, 2, 2, 84531, 2, 531160, 2, 4738, 2, 3482830482, 2, 428195747108, 2

Offset: 2

Labos Elemer, Nov 05 2002

If a(n) = 2 then (n^n - 1) is in A013642. - Bernard Schott, Sep 25 2019

Cf. A003285, A048861.

Table[Length[Part[ContinuedFraction[Sqrt[ -1+u^u]], 2]], {u, 2, 15}]

a(n) = A003285(A048861(n)). - Michel Marcus, Sep 27 2019

a(17)-a(18) from Chai Wah Wu, Sep 24 2019

a(19)-a(20) from Chai Wah Wu, Sep 27 2019

A104132 Largest prime factor of pip(n)^pip(n)-1 where pip(n) is the n-th prime-indexed prime.

Original entry on oeis.org

13, 71, 1806113, 2699538733, 568972471024107865287021434301977158534824481, 5926187589691497537793497756719

Offset: 1

Cino Hilliard, Mar 06 2005

Amiram Eldar, Table of n, a(n) for n = 1..10

Cf. A006450, A006530, A006486, A048861, A104131, A214812.

lpf[n_]:=Module[{p=Prime[Prime[n]]},FactorInteger[p^p-1][[-1,1]]]; Array[lpf,6] (* Harvey P. Dale, Nov 09 2017 *)

piptopipm1(n) = { local(x, y); for(x=1, n, y=pip(x)^pip(x)-1; print1(bdiv(y)", "); ) }
pip(n) = { return(prime(prime(n))) }
bdiv(n) = { local(x); x=ifactor(n); return(x[length(x)]) }
ifactor(n, m=0) = { local(f, j, k, flist); flist=[]; f=Vec(factor(n, m)); for(j=1, length(f[1]), for(k = 1, f[2][j], flist = concat(flist, f[1][j]) ); ); return(flist) }

a(n) = A006530(A048861(A006450(n))). - Amiram Eldar, May 23 2020

a(6) corrected by Harvey P. Dale, Nov 09 2017

A128542 a(n) = ((2n)^(2n) - 1)/((2n+1)*(2n-1)).

Original entry on oeis.org

0, 1, 17, 1333, 266305, 101010101, 62350352785, 56984650387477, 72340172838076673, 121815504877079063701, 262801002506265664160401, 706890015246831381773595701, 2319540481478754999041880822337, 9120177155862455275254332279111413

Offset: 0

Alexander Adamchuk, May 08 2007

nonn

p divides a(p-1) for prime p>3.

G. C. Greubel, Table of n, a(n) for n = 0..190

Cf. A048861 = n^n - 1.

Cf. A023037, A089815.

Concatenation([0], List([1..20], n-> ((2*n)^(2*n)-1)/(4*n^2 -1) )); # G. C. Greubel, Jul 11 2019

[0] cat [((2*n)^(2*n)-1)/(4*n^2 -1): n in [1..20]]; // G. C. Greubel, Jul 11 2019

Join[{0}, Table[((2n)^(2n)-1)/(4n^2-1),{n,1,20}]]

A128542(n)=((n+=n)^n-1)/(n^2-1) \\ M. F. Hasler, Oct 31 2014

[0]+[((2*n)^(2*n)-1)/(4*n^2 -1) for n in (1..20)] # G. C. Greubel, Jul 11 2019

a(n) = ((2n)^(2n)-1)/((2n+1)*(2n-1)).
a(n) = A048861(2n)/((2n+1)*(2n-1)).
a(n) = A023037(2n)/(2n+1).
a(n) = A089815(2n-2).

a(0)=0 added by M. F. Hasler, Oct 31 2014

cp's OEIS Frontend

A116895 Least prime factor of n^n-1.

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A184966 Numbers k such that k^k - 1 is squarefree.

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Mathematica

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A366821 a(n) is phi(n^n-1) where phi is the Euler totient function.

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Maple

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

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Magma

Mathematica

A125556 Smallest prime p = n*m + 1 that divides m^m - 1 for some m > 1.

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PARI

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A225944 Numbers k such that prime(k) divides k^k - 1.

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Mathematica

Python

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A342930 Least positive number k such that n^n divides k*(k+1)/2.

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PARI

PARI

Formula

Extensions

A077098 Length of period of the continued fraction expansion of sqrt(-1+n^n).

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