A125135 -id:A125135 - 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

A088730 Numbers of the form p^p - 1, where p is a prime.

Original entry on oeis.org

3, 26, 3124, 823542, 285311670610, 302875106592252, 827240261886336764176, 1978419655660313589123978, 20880467999847912034355032910566, 2567686153161211134561828214731016126483468

Offset: 1

Cino Hilliard, Nov 23 2003

nonn

Sum of reciprocals = 0.3721161884983118696170302604..

			a(1) = 3 because the first prime is 2 and 2^2 - 1 = 3.
a(2) = 26 because the second prime is 3 and 3^3 - 1 = 26.
a(3) = 3124 because the fifth prime is 5 and 5^5 - 1 = 3124.

Vincenzo Librandi, Table of n, a(n) for n = 1..77

Cf. A051674, A088807, A125135 (factorizations).

[p^p-1: p in PrimesUpTo(20)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime[n]^Prime[n] - 1, {n, 10}] (* Alonso del Arte, May 22 2013 *)
#^#-1&/@Prime[Range[10]] (* Harvey P. Dale, Jun 10 2013 *)

a(n)={my(p=prime(n)); p^p-1;} \\ Joerg Arndt, May 27 2013

a(n) = A051674(n) - 1. - R. J. Mathar, Jul 15 2007

More terms from Ray Chandler, Feb 21 2004

A125137 a(n) = p^p + 1, where p = prime(n).

Original entry on oeis.org

5, 28, 3126, 823544, 285311670612, 302875106592254, 827240261886336764178, 1978419655660313589123980, 20880467999847912034355032910568, 2567686153161211134561828214731016126483470, 17069174130723235958610643029059314756044734432, 10555134955777783414078330085995832946127396083370199442518

Offset: 1

N. J. A. Sloane, Jan 21 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..77

See A125136 for factorizations. Cf. A088730, A125135.

Cf. A104131, A114846.

[p^p+1: p in PrimesUpTo(40)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime[n]^Prime[n] + 1, {n , 20}] (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = A051674(n)+1. - R. J. Mathar, Apr 23 2007

A214812 Largest prime factor of (p^p-1)/(p-1) where p = prime(n).

Original entry on oeis.org

3, 13, 71, 4733, 1806113, 1803647, 2699538733, 109912203092239643840221, 1920647391913, 549334763, 568972471024107865287021434301977158534824481, 41903425553544839998158239, 5926187589691497537793497756719, 19825223972382274003506149120708429799166030881820329892377241, 194707033016099228267068299180244011637

Offset: 1

N. J. A. Sloane, Jul 31 2012

nonn

Daniel Suteu, Table of n, a(n) for n = 1..33
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]

Cf. A001039, A088807, A125135, A212552, A214811.

FactorInteger[#][[-1,1]]&/@Table[(p^p-1)/(p-1),{p,Prime[Range[15]]}] (* Harvey P. Dale, Aug 27 2016 *)

a(n) = my(p=prime(n)); vecmax(factor((p^p-1)/(p-1))[,1]); \\ Daniel Suteu, May 26 2022

a(n) = A006530(A001039(n)). - Daniel Suteu, May 26 2022

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),", "))

A352711 The left Aurifeuillian factor of p^p - 1 for primes p congruent to 1 (mod 4).

Original entry on oeis.org

11, 1803647, 2699538733, 112663560435723374699, 6243610407478181159725577611, 67643278270835231300426724641533, 253382315888712050791030544452181354268272663, 133904013361225746608283522164245432446284642589451147, 4429523820749528526448423858097183945539957285504166342434080091097

Offset: 1

Patrick A. Thomas, Mar 30 2022

nonn

For prime factorizations of p^p - 1 see A125135.

Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - Bernard Schott, Nov 04 2022

			112663560435723374699 is the smaller Aurifeuillian factor of 29^29-1, and 29 is the 4th term of A002144, so a(4) = 112663560435723374699.

Patrick A. Thomas, Table of n, a(n) for n = 1..60
Calculators, Aurifeuillian LMs
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
Wikipedia, Léon-François-Antoine Aurifeuille.
Wikipedia, Aurifeuillean factorization.

Cf. A002144, A125135, A230376, A352732, A352400, A352401.

If R is (p^p-1)/(p-1), where p == 1 (mod 4) and p > 5, then an approximation of the left Aurifeuillian factor of R is (1/e) * sqrt(R/(1+z)), where z =
   2/(3p) + 28/(45p^2) + 1706/(2835p^3) if p=1,79,109,121,151  or 169 (mod 210),
   2/(3p) + 28/(45p^2) +   86/(2835p^3) if p=19,31,61,139,181  or 199 (mod 210),
   2/(3p) -  8/(45p^2) +  194/(2835p^3) if p=37,43,67,127,163  or 193 (mod 210),
   2/(3p) -  8/(45p^2) - 1426/(2835p^3) if p=13,73,97,103,157  or 187 (mod 210),
  -2/(3p) -  8/(45p^2) + 1426/(2835p^3) if p=23,53,107,113,137 or 197 (mod 210),
  -2/(3p) -  8/(45p^2) -  194/(2835p^3) if p=17,47,83,143,167  or 173 (mod 210),
  -2/(3p) + 28/(45p^2) -   86/(2835p^3) if p=11,29,71,149,179  or 191 (mod 210),
  -2/(3p) + 28/(45p^2) - 1706/(2835p^3) if p=41,59,89,101,131  or 209 (mod 210).

A352732 The right Aurifeuillian factor of p^p - 1, for primes p congruent to 1 (mod 4).

Original entry on oeis.org

71, 13993643, 19152352117, 813955076015309926319, 46959719470144429555105032871, 491873569944394295636860313807677, 1848593595048531176470116001230356265643249547, 1000403244183535565720394723140528028235711874491322863, 33027769942300819203735411144251223948236849608414254057770836237073

Offset: 1

Patrick A. Thomas, Mar 30 2022

nonn

For prime factorizations of p^p - 1 see A125135.

			813955076015309926319 is the larger Aurifeuillian factor of 29^29-1, and 29 is the 4th term of A002144, so a(4) = 813955076015309926319.

Patrick A. Thomas, Table of n, a(n) for n = 1..60
Calculators, Aurifeuillian LMs
Wikipedia, Aurifeuillean factorization.

Cf. A002144, A125135, A230378, A352711.

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

A214811 Triangle read by rows: row n lists prime factors of (p^p-1)/(p-1) where p = prime(n).

Original entry on oeis.org

3, 13, 11, 71, 29, 4733, 15797, 1806113, 53, 264031, 1803647, 10949, 1749233, 2699538733, 109912203092239643840221, 461, 1289, 831603031789, 1920647391913, 59, 16763, 84449, 2428577, 14111459, 58320973, 549334763, 568972471024107865287021434301977158534824481, 149, 1999, 7993, 16651, 17317, 10192715656759, 41903425553544839998158239

Offset: 1

N. J. A. Sloane, Jul 31 2012

			Triangle begins:
[3]
[13]
[11, 71]
[29, 4733]
[15797, 1806113]
[53, 264031, 1803647]
[10949, 1749233, 2699538733]
[109912203092239643840221]
[461, 1289, 831603031789, 1920647391913]
[59, 16763, 84449, 2428577, 14111459, 58320973, 549334763]
[568972471024107865287021434301977158534824481]
[149, 1999, 7993, 16651, 17317, 10192715656759, 41903425553544839998158239]
...

J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423. See Table 3.

Cf. A001039, A054767, A125135, A214810, A214812.

f:=proc(n) local i,t1,p,B,F;
p:=ithprime(n);
B:=(p^p-1)/(p-1);
F:=ifactors(B)[2];
lprint(n,p,B,F);
t1:=[seq(F[i][1],i=1..nops(F))];
sort(t1);
end;

A248843 Table read by rows in which row n lists divisors of (p^p-1)/(p-1) where p = prime(n).

Original entry on oeis.org

1, 3, 1, 13, 1, 11, 71, 781, 1, 29, 4733, 137257, 1, 15797, 1806113, 28531167061, 1, 53, 264031, 1803647, 13993643, 95593291, 476218721057, 25239592216021, 1, 10949, 1749233, 2699538733, 19152352117, 29557249587617, 4722122236541789

Offset: 1

Jean-François Alcover, Dec 03 2014

			Table begins:
  [1, 3],
  [1, 13],
  [1, 11, 71, 781],
  [1, 29, 4733, 137257],
  [1, 15797, 1806113, 28531167061],
  [1, 53, 264031, 1803647, 13993643, 95593291, 476218721057, 25239592216021],
  ...

Samuel S. Wagstaff, Aurifeuillian Factorizations and the Period of the Bell Numbers, Math. Comp. 65 (1996), 383-391.
Eric Weisstein's MathWorld, Bell Number
Wikipedia, Bell Number

Cf. A000110, A001039, A054767, A125135, A214810, A214811, A214812.

Table[p = Prime[n]; Divisors[(p^p - 1)/(p - 1)], {n, 1, 10}] // Flatten

cp's OEIS Frontend

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

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A088730 Numbers of the form p^p - 1, where p is a prime.

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A125137 a(n) = p^p + 1, where p = prime(n).

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A214812 Largest prime factor of (p^p-1)/(p-1) where p = prime(n).

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

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A352711 The left Aurifeuillian factor of p^p - 1 for primes p congruent to 1 (mod 4).

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A352732 The right Aurifeuillian factor of p^p - 1, for primes p congruent to 1 (mod 4).

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

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