A006486 -id:A006486 - cp's OEIS frontend

A007571 a(n) = largest prime factor of n^n + 1.

2, 5, 7, 257, 521, 97, 911, 673, 530713, 27961, 58367, 2227777, 79301, 176597, 142111, 67280421310721, 45957792327018709121, 33388093, 870542161121, 4406613081041681, 22864311556633, 73194743542229, 1522029233, 27250359649

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Tyler Busby, Table of n, a(n) for n = 1..148 (terms 1..123 from Hugo Pfoertner, terms 124..135 from Yurii Ivanov)
factordb, Status of 148^148+1 ... 167^167+1.

Cf. A006530, A006486, A014566, A056790.

[Max(PrimeDivisors(n^n+1)):n in [1..24]]; // Marius A. Burtea, Aug 24 2019

Table[ FactorInteger[ n^n + 1, FactorComplete -> True ] [ [ -1, 1 ] ], {n, 1, 25} ]

for(k=1, 24, my(x=factor(k^k+1), f=x[#x[, 1], 1]); print1(f,", ")) \\ Hugo Pfoertner, Aug 23 2019

a(n) = A006530(A014566(n)). - Michel Marcus, Aug 24 2019

A125135 Triangle read by rows in which row n lists prime factors of p^p - 1 where p = prime(n).

Original entry on oeis.org

3, 2, 13, 2, 2, 11, 71, 2, 3, 29, 4733, 2, 5, 15797, 1806113, 2, 2, 3, 53, 264031, 1803647, 2, 2, 2, 2, 10949, 1749233, 2699538733, 2, 3, 3, 109912203092239643840221, 2, 11, 461, 1289, 831603031789, 1920647391913

Offset: 1

N. J. A. Sloane, Jan 21 2007

			Triangle begins:
3;
2, 13;
2, 2, 11, 71;
2, 3, 29, 4733;
2, 5, 15797, 1806113;
2, 2, 3, 53, 264031, 1803647;
2, 2, 2, 2, 10949, 1749233, 2699538733;
2, 3, 3, 109912203092239643840221;
2, 11, 461, 1289, 831603031789, 1920647391913;
2, 2, 7, 59, 16763, 84449, 2428577, 14111459, 58320973, 549334763;
...
n=4: p=7, 7^7-1 = 823542 = 2*3*29*4733 gives row 4.

Sam Wagstaff, Factorizations of p^p - 1 for most p < 180

Cf. A006486, A088730, A125136, A212552, A214812.

for p in [ n : n in [1..100] | IsPrime(n) ] do "\nDoing p =", p; n := p^p -1; Factorisation(n); end for; // John Cannon

T:= n-> (p-> sort(map(i-> i[1]$i[2], ifactors(p^p-1)[2]))[])(ithprime(n)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 20 2022

A334167 a(n) is the number of divisors of n^n-1.

Original entry on oeis.org

2, 4, 8, 12, 16, 16, 96, 128, 48, 16, 256, 48, 32, 128, 128, 40, 512, 12, 2048, 768, 256, 64, 6144, 4096, 768, 512, 4096, 768, 24576, 16, 6144, 768, 8192, 1024, 262144, 1152, 256, 1024, 49152, 256, 65536, 64, 24576, 196608, 384, 32, 393216, 327680, 12288, 24576

Offset: 2

Todor Szimeonov, Apr 17 2020

nonn

25 values of the first 40 are powers of 2.

Max Alekseyev, Table of n, a(n) for n = 2..138

Cf. A000005, A048861, A006486, A116895, A344859.

a[n_] := DivisorSigma[0, n^n - 1]; a /@ Range[2, 45] (* Giovanni Resta, Apr 17 2020 *)

a(n) = numdiv(n^n-1); \\ Michel Marcus, Apr 17 2020

a(n) = A000005(A048861(n)).

More terms from Giovanni Resta, Apr 17 2020

A372228 a(n) is the largest prime factor of n^n + n.

Original entry on oeis.org

2, 3, 5, 13, 313, 101, 181, 5419, 21523361, 52579, 212601841, 57154490053, 815702161, 100621, 4454215139669, 4562284561, 52548582913, 1895634885375961, 211573, 2272727294381, 415710882920521, 9299179, 1853387306082786629, 22496867303759173834520497

Offset: 1

Tyler Busby, Apr 23 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..116

Cf. A066068, A006530, A006486, A007571, A372229.

Table[f = FactorInteger[n^n + n]; f[[Length[f]]][[1]], {n, 1, 25}] (* Vaclav Kotesovec, Apr 26 2024 *)

from sympy import primefactors
def A372228(n): return max(max(primefactors(n),default=1),max(primefactors(n**(n-1)+1))) # Chai Wah Wu, Apr 27 2024

a(n) = A006530(A066068(n)).

A309941 Number of prime factors of n^n - 1, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 7, 8, 6, 4, 8, 6, 5, 7, 7, 7, 10, 4, 11, 10, 8, 6, 13, 13, 11, 9, 13, 10, 15, 4, 13, 12, 13, 10, 18, 11, 8, 10, 16, 9, 16, 6, 15, 18, 9, 5, 19, 20, 14, 15, 17, 8, 16, 12, 18, 10, 10, 5, 26, 8, 10, 14, 20, 19, 17, 9, 17, 12, 19, 7, 29, 15, 8, 11, 20, 13, 21, 8

Offset: 2

Hugo Pfoertner, Aug 24 2019

			a(3) = 2: 3^3 - 1 = 26 = 2 * 13.
a(5) = 4: 5^5 - 1 = 3124 = 2^2 * 11 * 71.

Amiram Eldar, Table of n, a(n) for n = 2..138 (terms 2..126 from Chai Wah Wu)
factordb, Factors of n^n-1.

Cf. A006486, A048861, A085723, A116895, A125135, A334167, A344870.

a[n_] := PrimeOmega[n^n - 1]; Array[a, 45, 2] (* Amiram Eldar, Jul 04 2024 *)

for(k=2, 50, print1(bigomega(k^k-1),", "))

A372229 a(n) is the largest prime factor of n^n - n.

Original entry on oeis.org

2, 3, 7, 13, 311, 43, 337, 193, 333667, 13421, 266981089, 28393, 29914249171, 10678711, 1321, 184417, 7563707819165039903, 236377, 192696104561, 920421641, 12271836836138419, 39700406579747, 58769065453824529, 152587500001, 4315817869647001, 797161

Offset: 2

Tyler Busby, Apr 23 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 2..115

Cf. A061190, A006530, A006486, A007571, A372228.

pf := n -> NumberTheory:-PrimeFactors(n): a := n -> max(pf(n^n - n));
seq(a(n), n = 2..27);  # Peter Luschny, Apr 27 2024

Table[f = FactorInteger[n^n-n]; f[[Length[f]]][[1]], {n,2,25}] (* Vaclav Kotesovec, Apr 26 2024 *)

from sympy import primefactors
def A372229(n): return max(max(primefactors(n),default=1),max(primefactors(n**(n-1)-1),default=1)) # Chai Wah Wu, Apr 27 2024

a(n) = A006530(A061190(n)).

A116895 Least prime factor of n^n-1.

Original entry on oeis.org

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

A216487 Smallest prime factor of n^(2n) - 1 having the form k*n+1.

Original entry on oeis.org

3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 10949, 19, 108301, 41, 43, 23, 47, 73, 101, 53, 109, 29, 59, 31, 373, 257, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 89, 181, 47, 659, 97, 197, 101, 103, 53, 107, 109, 881, 113, 229, 59, 709, 61, 977

Offset: 2

Michel Lagneau, Sep 11 2012

nonn

The corresponding values of k are in A216506.

			a(7) = 29 because 7^14 - 1 = 2 ^ 4 * 3 * 29 * 113 * 911 * 4733 and the smallest prime divisor of the form k*n+1 is 29 = 4*7+1.

Amiram Eldar, Table of n, a(n) for n = 2..670

Cf. A006486, A007571, A187022, A187023, A216506.

Table[p=First/@FactorInteger[n^(2*n)-1]; Select[p, Mod[#1, n] == 1 &, 1][[1]], {n, 2, 41}]
a[n_] := Module[{m = n + 1}, While[!PrimeQ[m] || PowerMod[n, 2*n, m] != 1, m += n]; m]; Array[a, 100, 2] (* Amiram Eldar, May 17 2024 *)

a(n) = {my(m = n + 1); while(!isprime(m) || Mod(n, m)^(2*n) != 1, m += n); m;} \\ Amiram Eldar, May 17 2024

a(n) = Min{A187022(n), A187023(n)}.

Data corrected by Amiram Eldar, May 17 2024

A216506 Least number k such that k*n+1 is a prime dividing n^(2n) - 1.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 2, 2, 1, 2, 1, 4, 2, 2, 1, 644, 1, 5700, 2, 2, 1, 2, 3, 4, 2, 4, 1, 2, 1, 12, 8, 2, 3, 2, 1, 4, 5, 2, 1, 2, 1, 4, 2, 4, 1, 14, 2, 4, 2, 2, 1, 2, 2, 16, 2, 4, 1, 12, 1, 16, 273, 2, 3, 2, 1, 4, 2, 2, 1, 246, 1, 4, 2, 2, 16, 8, 1, 4, 15, 2, 1, 2, 4, 12

Offset: 2

Michel Lagneau, Sep 11 2012

nonn

The corresponding prime factors of n^(2n)-1 of the form k*n+1 is in A216487.

			a(7) = 4 because 7^14 - 1 = 2 ^ 4 * 3 * 29 * 113 * 911 * 4733 and the smallest prime divisor of the form k*n+1 is 29 = 4*7+1 => k = 4.

Amiram Eldar, Table of n, a(n) for n = 2..670

Cf. A006486, A007571, A187022, A187023, A216487.

Table[p=First/@FactorInteger[n^(2*n)-1]; (Select[p, Mod[#1, n] == 1 &, 1][[1]]-1)/n, {n, 2, 50}]
a[n_] := Module[{m = n + 1}, While[!PrimeQ[m] || PowerMod[n, 2*n, m] != 1, m += n]; (m - 1)/n]; Array[a, 100, 2] (* Amiram Eldar, May 17 2024 *)

a(n) = {my(m = n + 1); while(!isprime(m) || Mod(n, m)^(2*n) != 1, m += n); (m - 1)/n;} \\ Amiram Eldar, May 17 2024

Data corrected and extended by Amiram Eldar, May 17 2024

A319183 a(n) = phi(n^n - 1)/n where phi is A000010.

Original entry on oeis.org

1, 4, 32, 280, 5040, 37856, 829440, 15676416, 589032000, 10374307328, 388566097920, 7619466454080, 390751784579520, 11138729990400000, 575561351791902720, 24328359845627701248, 1640651748984970444800, 34709116765970413844280, 2459108342476800000000000

Offset: 2

Seiichi Manyama, Sep 12 2018

nonn

Main diagonal of the array T(n,k) = phi(n^k-1)/k for n > 1 and k > 1, which starts
    1,   2,   2,    6,     6,     18,     16, ... A011260
    2,   4,   8,   22,    48,    156,    320, ... A027385
    4,  12,  32,  120,   288,   1512,   4096, ... A027695
    4,  20,  48,  280,   720,   5580,  14976, ... A027741
   12,  56, 216, 1240,  5040,  31992, 139968, ... A295496
    8,  36, 160, 1120,  6048,  37856, 192000, ... A027743
   18, 144, 432, 5400, 23328, 254016, 829440, ... A027744

Seiichi Manyama, Table of n, a(n) for n = 2..50
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

A diagonal of A369291.

Cf. A000010, A006486, A027385.

Table[EulerPhi[n^n-1]/n,{n,20}] (* Harvey P. Dale, Aug 04 2020 *)

PARI
```
{a(n) = eulerphi(n^n-1)/n}
```

cp's OEIS Frontend

A007571 a(n) = largest prime factor of n^n + 1.

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A125135 Triangle read by rows in which row n lists prime factors of p^p - 1 where p = prime(n).

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A334167 a(n) is the number of divisors of n^n-1.

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A372228 a(n) is the largest prime factor of n^n + n.

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A309941 Number of prime factors of n^n - 1, counted with multiplicity.

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A372229 a(n) is the largest prime factor of n^n - n.

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A116895 Least prime factor of n^n-1.

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A216487 Smallest prime factor of n^(2n) - 1 having the form k*n+1.

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