A007571 -id:A007571 - cp's OEIS frontend

A006486 a(n) = largest prime factor of n^n - 1.

3, 13, 17, 71, 43, 4733, 241, 757, 9091, 1806113, 20593, 1803647, 8108731, 39225301, 6700417, 2699538733, 465841, 109912203092239643840221, 222361, 227633407, 285451051007, 1920647391913, 1134793633, 50150933101, 3574533119, 12557612956332313, 1100860153

Offset: 2

Dennis S. Kluk (mathemagician(AT)ameritech.net)

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

Table of n, a(n) for n = 2..138
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979.

Cf. A006530, A007571, A048861, A125135.

[Max(PrimeDivisors(n^n-1)):n in [2..28]]; // Marius A. Burtea, Aug 24 2019

Table[Max@Transpose[FactorInteger[n^n - 1]][[1]], {n, 2, 28}] (* Arkadiusz Wesolowski, Aug 06 2012 *)

for(k=2, 28, my(x=factor(k^k-1), f=x[#x[, 1], 1]); print1(f,", ")) \\ Hugo Pfoertner, Aug 23 2019

a(n) = A006530(A048861(n)). - Michel Marcus, Aug 24 2019

Corrected by T. D. Noe, Nov 14 2006
5 more terms from Arkadiusz Wesolowski, Aug 06 2012
Terms up to a(126) in b-file added by Sean A. Irvine, Apr 25 2017
Terms a(127)-a(138) in b-file added by Max Alekseyev, Aug 26 2021

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)).

A344859 a(n) is the number of divisors of n^n + 1.

Original entry on oeis.org

2, 2, 2, 6, 2, 8, 8, 16, 8, 16, 8, 96, 16, 32, 48, 160, 4, 12, 288, 48, 8, 64, 16, 512, 64, 128, 32, 3072, 64, 128, 1024, 384, 16, 2048, 64, 18432, 32, 128, 192, 512, 768, 64, 1024, 384, 256, 16384, 256, 2560, 64, 192, 1024, 3072, 32, 512, 16384, 4096, 128, 8192, 8192, 768, 4096, 256, 128, 1376256, 16

Offset: 0

Seiichi Manyama, May 31 2021

nonn

Tyler Busby, Table of n, a(n) for n = 0..148 (terms 0..123 from Max Alekseyev)
Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind

Cf. A000005, A014566, A062319, A064144, A121270, A334167.

Cf. A007571 (Lpf), A055385 (lpf).

a[0] = 2; a[n_] := DivisorSigma[0, n^n + 1]; Array[a, 45, 0] (* Amiram Eldar, May 31 2021 *)

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

a(n) = A000005(A014566(n)).

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)).

A085723 Number of prime divisors of n^n+1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 5, 3, 4, 3, 7, 4, 5, 6, 9, 2, 4, 9, 6, 3, 6, 4, 10, 6, 7, 5, 12, 6, 7, 10, 11, 4, 11, 6, 15, 5, 7, 8, 10, 10, 6, 10, 9, 8, 14, 8, 13, 6, 8, 10, 12, 5, 10, 14, 13, 7, 13, 13, 10, 12, 8, 7, 24, 4, 12, 8, 8, 7, 17, 10, 11, 12, 4, 8, 25, 7, 9, 14, 10, 5, 12, 7, 13, 8

Offset: 1

Jason Earls, Jul 20 2003

16^16+1 = 274177 * 67280421310721 is a semiprime. Where is the next?
a(73) >= 4. - Donovan Johnson, Sep 27 2010
According to factordb there are currently no other known candidates for semiprimes, with 781^781+1 being the largest fully factored number of this form. - Hugo Pfoertner, Aug 24 2019

			a(3) = 3: 3^3 + 1 = 28 = 2^2 * 7.
a(4) = 1: 4^4 + 1 = 257 is prime.
a(5) = 3: 5^5 + 1 = 3126 = 2 * 3 * 521.

Amiram Eldar, Table of n, a(n) for n = 1..148 (terms 1..123 from Chai Wah Wu)
factordb, Factors of n^n+1.

Cf. A001222, A007571, A014566, A055385, A121270, A309941, A344869.

for(k=1, 60, print1(bigomega(k^k+1),", ")) \\ Hugo Pfoertner, Aug 24 2019

a(n) = A001222(A014566(n)). - Amiram Eldar, Sep 27 2024

More terms from Ray G. Opao, Aug 25 2004
Corrected 8 existing terms and a(46)-a(72) from Donovan Johnson, Sep 27 2010
a(73)-a(84) added by Hugo Pfoertner, Aug 24 2019

A125136 Triangle read by rows in which row n gives list of prime factors of p^p + 1 where p = prime(n).

Original entry on oeis.org

5, 2, 2, 7, 2, 3, 521, 2, 2, 2, 113, 911, 2, 2, 3, 23, 89, 199, 58367, 2, 7, 13417, 20333, 79301, 2, 3, 3, 45957792327018709121, 2, 2, 5, 108301, 1049219, 870542161121, 2, 2, 2, 3, 47, 139, 1013, 1641281, 52626071, 1522029233, 2, 3, 5, 233, 6864997

Offset: 1

N. J. A. Sloane, Jan 21 2007

Product over the n-th row of the table is A051674(n) + 1. The number of elements in the n-th row is A115973(n). - R. J. Mathar, Jan 22 2007

(p + 1) divides p^p + 1 for odd prime p. - Alexander Adamchuk, Jan 22 2007

			Rows read
  5;
  2, 2, 7;
  2, 3, 521;
  2, 2, 2, 113, 911;
  2, 2, 3, 23, 89, 199, 58367;
  2, 7, 13417, 20333, 79301;
  2, 3, 3, 45957792327018709121;
  2, 2, 5, 108301, 1049219, 870542161121;
  2, 2, 2, 3, 47, 139, 1013, 1641281, 52626071, 1522029233;
  2, 3, 5, 233, 6864997, 9487923853, 5639663878716545087233;
  2, 2, 2, 2, 2, 373, 1613, 62869, 145577, 35789156484227, 2706690202468649;
  etc.

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

Cf. A088730, A125135.

Cf. A007571 = largest factor of n^n + 1.

pfs := proc(n) local ifs,a,e,b ; ifs := ifactors(n)[2] ; a := [] ; for b from 1 to nops(ifs) do for e from 1 to op(2,op(b,ifs)) do a := [op(a),op(1,op(b,ifs))] ; od ; od ; RETURN(a) ; end; A125136 := proc(nmax) local a,p,n,pp ; a := [] ; p := 2 ; while nops(a) < nmax do a := [op(a),op(pfs(p^p+1))] ; p := nextprime(p) ; od ; RETURN(a) ; end; A125136(40) ; # R. J. Mathar, Jan 22 2007

lpf[n_]:=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]; lpf/@(#^#+1&/@ Prime[Range[10]])//Flatten (* Harvey P. Dale, Oct 18 2020 *)

More terms from Alexander Adamchuk and R. J. Mathar, Jan 22 2007

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

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)).

cp's OEIS Frontend

A006486 a(n) = largest prime factor of n^n - 1.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A344859 a(n) is the number of divisors of n^n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A085723 Number of prime divisors of n^n+1 (counted with multiplicity).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A125136 Triangle read by rows in which row n gives list of prime factors of p^p + 1 where p = prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A116895 Least prime factor of n^n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI